If your allergy is this severe, you honestly don't stand a chance.

Hot pot itself won't generally contain egg products, but cross contamination (which is going to be your big problem anyway) can still be an issue. Some people also dip meat from the hot pot into bowls of raw egg, and will then use their chopsticks (now with traces of egg) to grab another bite from the pot. Honestly, having seen the inside of a good number of restaurant kitchens in China, even white rice isn't necessarily safe from having come in contact with a serving utensil or bowl also used at some point for something containing eggs.