Eggless Egg Replacement?

Hello -

Looking for products/whatever to replace eggs - mainly in baking. Roomie is allergic to eggs - can't eat any egg products and even some products that were just processed on the same machinery etc. With the holidays here we are looking for something that can be used in place of eggs. Was recommended to stay away from all eggs, not just chicken eggs.

Have heard rumors about using banana, apple sauce, etc. Has anyone done this or know more about the amounts used? Any vegans have any suggestions?

Thanks