My friend and her son are allergic to eggs and she shared this with me. I thought it would be a great way to get more veggies in my kids' diet and with the cost of eggs it helps save some money.
For every two (2) eggs substitute 1/2 cup pumpkin. This works really well in banana bread and cake mixes they stay super moist. I also tried it in pancakes and it works well. They have a very slight pumpkin taste, but tasted great and were super moist.
Source: Melanie
This page contains the following solutions.
My husband and I are vegetarians, and have found that one heaping tablespoon of sour cream equals one egg in any recipe. It works in any baking recipe I've tried (we cook a lot).
There are still cakes and cookies that you can make when eggs are not allowed in your diet. This page contains baking recipes without eggs.
Eggs are typically used as a binder in meatloaf. If you are eliminating eggs from your diet, there are ingredients you can substitute. This is a page about substitutes for egg in meatloaf.