I don't know your fear of beef and pork, but you need to include those in your menu. You can just serve chicken. The broader and scopier your meat choices, the more likely you are to come close to fulfilling your dog's nutritional needs. All meat is good meat. You can shy away because of the bone issue. As long as whatever you give your dog is 80% meat and 10% bone, it really doesn't matter what type of meat you serve up.
It's been proven that dogs who break teeth would have also very likely have broken them on rawhide sticks, wood, and other objects. Some dogs have teeth that are very striated and it is these striations the make for broken teeth. This is a genetic trait, and it doesn't take bones to break those teeth, it can happen on virtually any hard object ( so that you know).
The part of the egg that is nutritional is the yolk. It contains valuable fat, and whether it's cooked or not won't make much of a difference. Fat is fat whether raw or cooked.