The short answer is: it depends–it depends on what you mean by vegetarian and what you mean by healthy, and on which life stage and which part of the world you’re talking about.
The vegetarian label can encompass diets that include fish, eggs, and dairy. People who eat less processed and red meat tend to live longer, so on that count, being vegetarian will probably give you a better shot at a longer life. Not that many people have been pure vegans, abstaining from any animal products, so there isn’t a lot of scientific data on whether such diets promote longer lives. The effects of a vegan diet that includes a lot of sugar and processed junk food and that lack vitamin B12 are surely deleterious, so the details matter.
But what exactly do we mean by healthy, anyway? For some people, probably most, healthy means having a longer life, but a vegetarian diet probably won’t help a weightlifter or Sumo wrestler win a title. The cholesterol from animal foods is plausibly linked to increased sex drive as well, no small consideration. Another point to keep in mind is that what’s longevity-boosting for a youngster may be quite different from what’s longevity-boosting for an elderly person. One recent study has pointed out that people over the age of 65 who consume more protein (from either animal or plant foods) may have relatively longer lives, perhaps due to the benefits of protein consumption in maintaining weight and IGF-1 hormone levels in frail elderly people.
A final consideration is what part of the world we’re looking at. People in some places of the world, such as the Arctic Inuit and cattle-herders in East Africa, biologically and culturally evolved with diets that consisted largely of animal foods, so to ask them to switch to a vegetarian diet seems questionable. That said, in most parts of the world, meat was scarce in traditional diets, until rises in income allowed greater exploitation of artificial fertilizers to feed livestock, bringing more meat to the table–at least for the well-off. Raising large livestock can have negative consequences for the environment in terms of greenhouse gas emissions, water use, and water pollution, so from an ecological standpoint, we’re better off going back to low-meat traditional diets and supplementing our protein cravings with plant and insect substitutes, along with locally-sourced game. We can grant our elders their deserved meat-fest indulgences.