Answer:
In English, it's customary to define people by what they do eat, rather than what they do not. For example most people are considered omnivores. Omni from Latin means "all, every", and vore, from the Latin word voro means "eating, gaining sustenance." So omnivores are "all eaters"; they eat anything or everything.
Then we have vegetarians (from the combination of vegetarian and agrarian) who eat only vegetables. Pescetarians, also called pesco-vegetarians (pesco is from the Italian word for fish pesce) are people who eat fish and vegetables but no red meat. There are lacto-vegetarians (a.k.a. lactotarians), those who consume dairy products in addition to veggies, and ovo-vegetarians-or ovotarians-who consume eggs but no dairy, fish, or red meat. Finally there are lacto-ovo vegetarians, those who eat dairy products, eggs and vegetables. Those are the most common "arians."
A term for a person who doesn't eat eggs would be something along the lines of "anti-ovotarian." Anti, which means "a person who is opposed to a particular practice" added to the word ovotarian, results in the meaning "person who opposes the practice of eating eggs."