Udis are not egg free.
We just came back yesterday and they had GNI buns at a few places (apparently they are egg free) but I didn't try them because I don't try anything new like that on vacation unless I have personally verified with the company.
Otherwise, I just ate everything without bread. I had cheesesteak at the plaza 2x without a roll, a burger at Sci Fi without a bun, and a burger at POP without a bun. No big deal to me because i am used to not eating bread when out of the house, but obviously it matter more to other people.
I am anaphylactic to peanuts, tree nuts, eggs and shellfish and also allergic to sesame, wheat, oats, coconut, broccoli, all raw vegetables, and all raw fruits (except grapes) and soy (except soy lechitin and soybean oil) and I just ate Amazingly well for the 6 days we were there.