Yes, it is possible to be allergic to the protein in egg whites, egg yolks, or the whole egg. Eggs are one of the top eight allergens and in the US are required to be listed on labels clearly for that reason.
In the health food store there is something called egg replacer by Ener-G foods. I believe it is made primarily out of tapioca starch. You mix it with water and use it to replace eggs. I've not baked with it myself, but I've heard good things from others who have. That being said, I think it would work better in brownies or quick bread than in bread pudding
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Re: I have A question regarding eggs please help
