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.