You have to eat meat and animal products to be healthy
This is false. There’s not anything humans need to survive which can only be found in animal products. One common worry is that it's impossible to get enough protein on a vegan diet, but there are plenty of plants containing proteins.
However, many plants are not complete proteins. Proteins are made of amino acids. The body can synthesize some amino acids , but we need to get 9 amino acids through our diets. If a food product contains all the 9 essential amino acids, we call them a complete protein. While many plants are not complete proteins, combinations of plants fill your dietary needs.  For example rice is low in lysine, however if you combine it with a legume like beans or lentils which are high in lysine, you can obtain all the 9 essential amino acids. However, there are also some select plants that have relatively high concentrations of all essential amino acids. Examples include, quinoa, and soy products such as tofu and tempeh. For more information see "Protein and vegetarian diets by Kate Marsh et al. 2013".
Another common worry is that you don't get enough vitamin B12. However even vitamin B12 which you normally get en passant from eating animals can be taken directly in supplement form. 
The “Vegetarian, vegan diets and multiple health outcomes Monica Dinu et al. 2017” systematic review, which is the highest form of scientific evidence, concludes after examining 86 cross sectional and 10 cohort studies on a variety of health-factors:
This comprehensive meta-analysis reports a significant protective effect of a vegetarian diet versus the incidence and/or mortality from ischemic heart disease (−25%) and incidence from total cancer (−8%). Vegan diet conferred a significant reduced risk (−15%) of incidence from total cancer.
And none of the 54 identified studies citing this one has refuted its claims. Another review “Relation of Vegetarian Dietary Patterns With Major Cardiovascular Outcomes” by Andrea J. Glenn et al. 2019 cites the study and finds their results to be consistent with the previous studies:
Our results are consistent with systematic-reviews and meta-analyses of prospective cohorts previously conducted in this area, where vegetarian dietary patterns were associated with a 25% reduced risk of CHD mortality. The same study also found that vegetarian dietary patterns were not significantly associated with CVD and stroke mortality.
However, the review “Veganism, vegetarianism, bone mineral density, and fracture risk” Isabel Iguacel et al. 2018 which looks at bone fractures finds
Twenty studies including 37 134 participants met the inclusion criteria. Compared with omnivores, vegetarians and vegans had lower BMD at the femoral neck and lumbar spine and vegans also had higher fracture rates.
Vegetarian and vegan diets should be planned to avoid negative consequences on bone health.
While looking at these studies, one should keep in mind that they are not long term randomized controlled trials, and since they are correlation based, one should not ignore the possibility that people who eat plant-based diets may generally be more health conscious than the average population, and may engage in fewer risk behaviors which could explain some of the apparent benefits from eating plant-based. However, one can reasonably conclude from this that you don’t have to eat meat in order to be healthy even if it requires more planning, and a change of habits.
Eating meat is inherently unhealthy
While red and especially processed meat has been associated with health risks, and many countries are starting to recommend limiting the intake of those kinds of meat, systematic reviews find insufficient evidence to recommend against eating white meat such as chicken and fish on the basis of health. Moreover, nations with high life-expectancy often also have a high consumption of fish, and plants.
The WHO recently classified processed meat as Group 1 carcinogenic to humans, and red meat as Group 2A carcinogenic to humans. They write
This recommendation was based on epidemiological studies suggesting that small increases in the risk of several cancers may be associated with high consumption of red meat or processed meat. Although these risks are small, they could be important for public health because many people worldwide eat meat and meat consumption is increasing in low- and middle-income countries.
The Group 1 and 2A classifications only refer to strength of evidence (strong and limited respectively in this case), and not strength of effect as some pro-vegan propaganda sources imply. The WHO specifically addresses these sources when they respond to “Does it mean that consumption of processed meat is as carcinogenic as tobacco smoking and asbestos?”
No, processed meat has been classified in the same category as causes of cancer such as tobacco smoking and asbestos (IARC Group 1, carcinogenic to humans), but this does NOT mean that they are all equally dangerous. The IARC classifications describe the strength of the scientific evidence about an agent being a cause of cancer, rather than assessing the level of risk.
The Global Burden of Disease Project estimates around 34 000 cancer deaths per year worldwide from diets high in processed meat, and if the links are shown to be causal 50 000 worldwide cancer deaths per year from red meat consumption compared to 1 million cancer deaths per year from smoking tobacco globally.
Some cultures live by eating meat more or less exclusively, and relatively healthy lives at that. They do so by eating the whole animal instead of just the lean muscle tissue we are accustomed to eating. For what a ‘westernized’ pure carnivore diet might look light, you may want to read the Guardian article “My carnivore diet: what I learned from eating only beef, salt and water”.
Veganism is always healthy
Just as you can be a healthy vegan, you can also be an unhealthy vegan if you don’t watch your diet. Oreos are vegan. A diet consisting of only Oreos is vegan, but clearly not healthy. So no; a diet being vegan does not magically mean it is a healthy diet.
Staying healthy as a vegan may or may not be more difficult than staying healthy as non-vegan, but no diet is a guarantee of health. Even a ‘healthy’ diet if consumed in excess is unhealthy. However, vegan diets naturally being rich in plants naturally lends itself to be more healthy than a diet that is rich in red and processed meats.
Veganism is bad for the climate
Systematic reviews consistently conclude that vegan diets reduce greenhouse gas (GHC) emissions compared to other diets . For example, the review “Environmental impact of dietary change: a systematic review” by E. Hallström et al. 2015 which included 14 studies assessing GHG emissions and land use for 49 dietary scenarios. The study found that dietary changes could reduce GHG emissions and land use by up to 50%, and the primary factors were amount and type of meat consumed with white meat and fish being more environmentally friendly than red and processed meat.
The results suggest that dietary change, in areas with affluent diet, could play an important role in reaching environmental goals, with up to 50% potential to reduce GHG emissions and land use demand associated with the current diet.
A United Nations Environment Programme report published in 2010 concluded:
A substantial reduction of environmental impacts would only be possible with a substantial worldwide diet change, away from animal products.
So it would appear that meat is a big contributor to climate change. This makes sense qualitatively as well since animals are entropy systems, so you require (much) more calories in than you get calories out. In fact, most of the agriculture land use goes to feeding animals instead of feeding humans directly.
Even looking at relatively high footprint plants such as soya and avocados, their footprints per gram, and per calorie is somewhere between 10 and 30 times less than that of beef.
It should be noted that a vegan diet is not the only way to reduce GHG emissions - a relative reduction of meat consumption will still contribute the relative portion to the reduction. It’s not a binary choice. Limiting meat consumption, and eating less red meat will help still reduce the carbon footprint.
Humans are made to eat meat or made to not eat meat
That is a nonsense argument humans are not made to do anything. If humans can do it, it’s natural, and it’s something humans are made for. Adaptations for eating meat probably came quite late in the evolutionary history during a time when access to plants weren’t as easily accessible. So humans if anything are made to eat both meat and plants.
But that’s entirely beside the point; evolution doesn’t have direction. There’s no should, or natural. There’s only can and cannot do. Humans can eat meat, and humans can eat plants, so that’s all that matters. Appeals to nature do not mean something is better. Much of what we do today is ‘unnatural’ - vaccines, and antibiotics are ‘unnatural’ - we didn’t have those when we were foragers, but they are keeping us alive today, and they are preventing a lot of harm.
It’s immoral to kill animals
Moral arguments are tricky due to the problems with absolute moralism. However, modern science indicate that many animals, including all mammals and most birds, have a emotional system very similar if not identical to that of humans.
Even more so for cows and pigs who are highly social animals and show symptoms of depression, PTSD, trauma, and stress very similarly to humans. So animals in factory farming conditions are probably feeling much like you’d feel if you’re in those conditions.
So if you don’t want to hurt people because you want to limit suffering, one might argue that you might want to look into extending the same empathy to animals with similar emotional systems. Whether that’s by ensuring humane living conditions for animals, or by not eating animals altogether, or if you don’t consider animals to have the same rights or whatever is largely a non-empirical matter as such there isn’t much to ‘debunk’.
Eating meat is sustainable
From the sources mentioned in this answer, we have seen that meat production is a major contributor to the climate crisis, and with low and medium income countries eating increasingly more meat, current meat consumption rates seem unlikely to be sustainable. Moreover, if you think animals should be able to walk around and graze outside, and not be stuck in factory-farming conditions, then we physically do not have enough land to sustain current and future meat consumption levels.
However, it’s not necessary to completely cut out meat consumption, a (drastic) reduction will go a long way to make meat production sustainable both ethically and for the environment.
The high use of antibiotics in factory farming pose a global health problem
The high use of antibiotics in meat production is problematic in two ways. Firstly, high use of antibiotics increase the risk of multi-resistant-bacteria, and contributes to the antibiotic crisis Moreover, when we need to use this level antibiotics, there’s an increased risk of new human illnesses.
: Proteins in human nutrition:
Amino acids in food and blood from Wikimedia by Jakob Suckale based on data from Julie Schmidt 2016.
We see that both vegan and meat-eaters have the about the same concentration of amino acids in their diets, and in their blood. We further see that vegans (eating balanced diets) don’t appear to be in any shortage of amino acids compared to regular non-vegan diets.
Moreover, we notice that the blood concentrations of the 9 essential amino acids that is amino acids which the body cannot produce itself in sufficient quantities (His, Ile, Leu, Lys, Met, Phe, Thr, Trp and Val) are very comparable in vegans and non-vegans.
: Vitamin B12 comes from a bacteria earth bla.bla.bla. Animals get it from either eating small amounts of dirt when eating their food or by getting it synthetically in pill form.
: The systematic review “Which Diet Has the Least Environmental Impact on Our Planet? A Systematic Review of Vegan, Vegetarian and Omnivorous Diets” by Bingli Clark Chai et al. 2019 which included 16 studies and 18 other reviews concludes
Results from our review suggest that the vegan diet is the optimal diet for the environment because, out of all the compared diets, its production results in the lowest level of GHG emissions.
the reviewed studies indicate the possibility of achieving the same environmental impact as that of the vegan diet, without excluding the meat and dairy food groups, but rather, by reducing them substantially.
Since the paper was published in MDPI - a journal which has previously published articles of dubious quality, and the due to the methodical limitations of the paper as well as the questionable scholarship and conclusions, I have chosen to not include this study in the main text. However, the paper has been peer reviewed, and the method is sound albeit limited which is why I have chosen to include it in the notes. The lack of effect size, and opaque uncertainty estimations should warrant skepticism of the strong wording used in the conclusions.