I would carry bilingual cards but with a stronger warning, as in "If I eat anything with eggs, I will die." I am serious about this. The Chinese definitely don't understand the implications of any serious food allergy, so hitting them over the head with the message is perfectly OK.

I actually find eggs one of the easier things to avoid in China. I've been doing it successfully for 10+ years. I am not allergic, but hate all eggs, all forms, with the passion of a thousand burning nuns. Many, many Chinese dishes have no egg at all in them--the things to particularly avoid would be fried rice dishes and some noodle dishes including all egg noodle types. And of course, obvious egg-based dishes and soups with egg threads in them. Be particularly careful of soups. Also be careful of dumplings, which can often contain pieces of egg.

I find that most meat and vegetable dishes, have no egg whatsoever. Ditto for doufu (tofu) dishes. Order dishes that are stir-fried, not deep fried, as the batter coating deep fried food may contain egg products. Steamed white rice is always safe.

There is no way, unfortunately, to deal with the issue of potentially contaminated utensils. Although I think the danger of secondary contamination may be a bit overblown. You should probably carry medication, an epi-pen, or whatever you must to combat inadvertent encounters arising from this sort of thing.

Dining in Western restaurants/hotel restaurants is no guarantee. Do not be lulled into a false sense of safety by relying on this gambit.