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Answer
As any good food science book will tell you, eggs are a wholly unique and miraculous food product that contribute many things to a recipe. They are particularly useful for their versatile proteins and their emulsifying abilities. You may be able to use an egg substitute (Egg Beaters, for instance) but do not attempt to use condensed milk or applesauce as a substitute. Your recipe will be doomed.
here is the whole article...
http://wiki.answers.com/Q/What_can_be_u ... ing_a_cakeyou're going to have to leave the house to drive to church anyway, so you may as well leave to get the eggs. sounds kind of old fashioned but if you're friendly with your neighbors, you could always ask to borrow a couple. good luck!
chi mangia bene, viva bene.
amen.