The health arguments are nonsense. Whether you are a meat eater or a vegetarian it takes effort to eat healthily. Either way you can have any two of 'tasty' 'healthy' or 'convenient' but not all three.
If you are veggie you have to be careful not to end up eating loads of (high fat) cheese, or (if vegan) you have to be careful to get all the trace nutrients, iron, omega, vitamins etc. If you are a meat-eater you have to be careful not to overdo the red meat and remember to eat lots of veg as well!
The argument for vegetarianism is a moral one, not a health one. Also the environmental argument is quite strong (producing meat takes a lot more resources than producing non-meat food).
It annoys me the way people like PETA will tell lies about the health risks of meat or milk etc. They should just stick to the moral argument (and let people make up their own minds on that basis). Exaggerating the health issues just loses them credibility.
Having said all that, I'm not a veggie, I've tried and failed several times.