I am thinking about starting a vegan diet.
Have fun with that. Veganism is totally abstaining from any animal products. 
I've done the vegetarian thing before, but never full vegan. I'm doing it for health reasons. Are there other reasons to become vegan?
None that I can think of. You basically can't even eat eggs or drink milk. 
Yeah, it'll be a big change. But the benefits will outweigh the negatives, hopefully.
Supposedly it helps reduce the risk of heart disease if you plan well. 
Are there people that shouldn't eat a vegan diet?
Yeah, children or teenagers are not recommended to eat a vegan diet.  Also pregnant women. 
I can understand children, but pregnant women surprises me.
not only follow a vegan diet but extend the philosophy into other areas of their lives, and oppose the use of animals for any purpose. Another term is "environmental veganism", which refers to the avoidance of animal products on the premise that the industrial farming of animals is environmentally damaging and unsustainable.
Well-planned vegan diets are regarded as appropriate for all stages of life, including during infancy and pregnancy, by the American Academy of Nutrition and Dietetics, Dietitians of Canada, Australian National Health and Medical Research Council
List of vegans
Veganism involves observing a vegan diet, which is a diet that includes no animals or animal products of any kind. It can extend to an entire lifestyle that precludes animal products from all aspects of life. Besides food, vegans may also choose to avoid clothes that use products such as leather and fur, cosmetics that have been tested on animals, and blood sports such as bullfighting and fox hunting. People who adopt veganism for ethical reasons may avoid or boycott all products or activities whose production or undertaking is