I am almost positive that they use PWE (processed whole eggs) that come in a milk type carton. Those eggs are pasteurized so they should be OK. I am in Florida but not near Disney, we still have eggs in the grocery stores here. That being said, it may just depend on where they get their eggs from. I am sure though, that if they were effected, they have found another place to get them.
Upon further thinking, they have to have eggs at Disney. Almost all the baked goods are made on property and from scratch. Desserts, cakes brownies and such, use eggs.