This article suggests that vegetarians are more health-conscious as a group than the average joe, making them healthier as a group--so it's a chicken & egg argument:

Proc Nutr Soc. 2006 Feb;65(1):35-41.
Health effects of vegetarian and vegan diets.

Key TJ, Appleby PN, Rosell MS.
Cancer Research UK Epidemiology Unit, University of Oxford, Richard Doll Building. tim.key@ceu.ox.ac.uk
Vegetarian diets do not contain meat, poultry or fish; vegan diets further exclude dairy products and eggs. Vegetarian and vegan diets can vary widely, but the empirical evidence largely relates to the nutritional content and health effects of the average diet of well-educated vegetarians living in Western countries, together with some information on vegetarians in non-Western countries. In general, vegetarian diets provide relatively large amounts of cereals, pulses, nuts, fruits and vegetables. In terms of nutrients, vegetarian diets are usually rich in carbohydrates, n-6 fatty acids, dietary fibre, carotenoids, folic acid, vitamin C, vitamin E and Mg, and relatively low in protein, saturated fat, long-chain n-3 fatty acids, retinol, vitamin B(12) and Zn; vegans may have particularly low intakes of vitamin B(12) and low intakes of Ca. Cross-sectional studies of vegetarians and vegans have shown that on average they have a relatively low BMI and a low plasma cholesterol concentration; recent studies have also shown higher plasma homocysteine concentrations than in non-vegetarians. Cohort studies of vegetarians have shown a moderate reduction in mortality from IHD but little difference in other major causes of death or all-cause mortality in comparison with health-conscious non-vegetarians from the same population. Studies of cancer have not shown clear differences in cancer rates between vegetarians and non-vegetarians. More data are needed, particularly on the health of vegans and on the possible impacts on health of low intakes of long-chain n-3 fatty acids and vitamin B(12). Overall, the data suggest that the health of Western vegetarians is good and similar to that of comparable non-vegetarians.

For specific health related issues (in this case Rheumatoid arthritis) a Vegan diet is actually recommended over all other diets:

- Kjeldsen-Kragh J. Rheumatoid arthritis treated with vegetarian diets. Am J Clin Nutr. 1999;70:594S-600S.
- Kjeldsen-Kragh J, Haugen M, Borchgrevink CF, et al. Controlled trial of fasting and one-year vegetarian diet in rheumatoid arthritis. Lancet. 1991;338:899,902
- Nenonen M, Helve T, Hanninen O. Effects of uncooked vegan food "living food" on rheumatoid arthritis, a three-month controlled and randomised study. Am J Clin Nutr. 1992;56:762.

There are a few small studies showing Vegan diets improve other conditions, but nothing as positive as the improvements with arthritis. Cancer is still up in the air--it seems that most studies show that eating lots of organic vegetables are of benefit whether meat is eaten or not. Typically, smaller, organic growers can produce healthier produce because the soil is more fertile and plant varieties are not limited to those that produce the most per acre (see In Defense of Food, Pollan, M.).