Okay, I need a baking genius to tell me how to do that without any eggs. . . .please!!

A baking genius I am not, but you can always substitute applesauce or mashed banana for eggs, just use the same amount (1 egg is about 1/4 cup). This takes away some of the fluffiness, so you might want to add a tiny pinch of baking powder or baking soda to help lift the cake.

I just Googled for other possible substitutes and read that regular (NOT diet) Coke or Pepsi also works! Again, using the same amount of soda as you would eggs and adding it last to preserve the bubbles. This way, you also don't need to add a leavening.

I've heard of substituting applesauce for oil before, but never applesauce for eggs.

Behold this Wonder! Applesauce!

It also just occurred to me that you could use Guinness or any other dark, chocolately stout with presumably delicious results.