I agree with the suggestion for applesauce.
I too have a son with an egg allergy and I have been researching this so he can have cupcakes for birthday parties, etc.
At the following website, I found substitutions:
http://www.thecookinginn.com/eggsub.html
I picked two substitutions and I baked cupcakes last night using a mix that required 3 eggs:
1 egg = 2 TBsp water + 1 TBsp oil + 2 tsp baking powder
1 egg = ¼ cup applesauce
I used applesauce for the first 2 eggs and the water/oil/powder for the 3rd egg. My cupcakes are VERY moist (but not gluey like some eggless cake reviews I have seen) and there is a hint of apple taste (but I like that) My 2 year old just gobbled one down and I did too.
I plan on looking for the egg replacement powder and trying that too!! The good news is... it looks like there are options. :-) I figure that I will make cupcakes and freeze them, and then pull one out of the freezer when my son goes to a birthday party.