Welcome to FlyerTalk!

You should memorise / print out these phrases:

我不能吃鸡蛋，我对鸡蛋过敏 // wo3 bu4 neng2 chi1 ji1dan4, wo3 dui4 ji1dan4 guo4min3 // I cannot eat eggs; I am allergic to them.
有什么菜不含有鸡蛋？ // you3 shen2me cai4 bu4 han2you3 ji1dan4 // What food do you have that does not contain eggs?

There are companies that sell little cards with these kind of things written on them, but they'll almost certainly just be a variation on what I've put above. If you show/tell this to restaurants, they should understand what you're saying, but whether they understand the implications is uncertain. Most reputable places should be good about telling you which menu items actually contain eggs, but all bets are off when it comes to avoiding all contamination with egg products.

As coplatflyer notes, food allergies are not a well understood concept in China, even in nicer (including Western) restaurants. Often, you may be interpreted as not wanting to eat something rather than not being able to. While people will generally make an effort, a lack of familiarity with these issues means that attempts will often be ineffectual. For instance, telling a Chinese person that you are allergic to peanuts will generally ensure that no actual peanuts end up in your food, many will not realise that a peanut allergy includes peanut oil.