I recently made a loaf of corn bread and realized halfway through that I was fresh out of eggs! This worked perfectly: 1 tbsp Flax Seed meal + 3 tbsp water, mixed together, then added. It binds easy cakes/loaves together nicely.
In your case, you won't even notice the flax seed flakes since the banana bread is naturally brown & speckled!

More banana would also work. The equivalent is 1/4 cup.

Oh, and pump up the baking powder in the recipe about 1/4 tsp to replace the egg which acts as a structure to aid for the rising of the bread.