Ahh, you should have talked to vegans instead. They don't use any animal products, no eggs, dairy, honey, etc.
Vegetarians- well it depends on what kind of vegetarian. Some do dairy,no eggs. Some do eggs, no dairy. Some are OK with both eggs and dairy but no flesh. Some do fish, most don't (fish- cold blooded).
It's very confusing if you try to lump all the non-meat eaters together. It made trying to order pizzas for working lunches almost impossible!
No animal products of any kind (not even to wear such as fur or leather), but fruit, nuts and veggies ok = vegan
No meat, eggs, or milk, but fur/leather ok = vegetarian
No meat or eggs, but milk products ok = lacto-vegetarian
No meat or milk products, but eggs ok = ovo-vegetarian
No meat, but eggs and milk ok = ovo-lacto-vegetarian
No warm blooded-animal flesh, but fish ok = pescetarian
No animal products or veggies, only fruit that has ripened and fallen from the tree naturally = hungry (er, fruititarian)
And, the most recent addition to the list, people who eat meat some days, but not every day = flexitarian (that would be moi)
I wonder if those who now insist on staying away from gluten are glutetarians? Or, unless they have specific allergies or celiac disease and are just following a trend, fool-itarians? I'm sure they'll soon come up with a name for it.