I have a World War II recipe for oatmeal chocolate chip cookies that uses
water and baking soda instead of eggs.
Eggs were in short supply then (I think) but this works well. It uses 1 tsp
of baking soda in 1/4 cup of boiling water. I don't know if this is
equivalent to one or two eggs. I haven't tried this in other recipes so I
don't know if it's a universal substitute but it's fine in the cookies.
I've made the recipe with both eggs and the water/baking soda substitute
and they come out the same.