My DD6 has a serious egg allergy so I always bake egg free. My go to replacement is:
1 egg =
1 1/2 tablespoon oil
1 1/2 tablespoon water
1 teaspoon baking powder
Make it when it is time to add the egg, give it a whisk until it is foamy then pour it in.
The baking powder needs to be fairly fresh but it works well in most recipes calling for 1-2 eggs