We have this same situation for my middle son. And it happens every time he has something that has alot of egg in it. ((ie french toast, straight eggs, etc)) I talked to his dr and he said if he isn't breaking out in hives or have a fever or swelling in his throat, it's a mild allergy and it would be very traumatic to make him go through allergy testing, since the best way, in his opinion, is the one where they do a bunch of needles in his skin. He said for now, just avoid the eggs.
I'd ask, has he had any issues with vaccines? If he has had reactions, you may want to get an allergy test done, to be safe.
Good luck, it's so painful to watch them get sick like that. :(