I greatly prefer brains rather than eggs.  I tried tanning a deer skin with eggs once, and it never softened up.  Don't know why, but it was as though the eggs created a layer in the skin that was impenetrable.  I then tried braining it and  the brains never got into the skin because the eggs created a boundary.  I never got the skin soft.  I used the entire egg, maybe I should have just used the yolks.  
I've heard of others using eggs with good results, but it just didn't work well for me.  I only use brains now.