What is the binding agent in eggs?
Egg whites contain a significant amount of protein, which is an excellent binder when cooked. This protein is what makes a cooked egg white somewhat rubbery.

Egg yolks contain lecithin, which is an emulsifier. The lecithin in an egg yolk is what combines with oil to make the emulsion we know as mayonnaise.

The white (with its protein) and yolk (with its lecithin and fat) are both critical to the chemistry when baking. If you eliminate just the yolk, you may get a tough and flat result. Substitutes like silken tofu, yogurt, applesauce, or mashed banana can all be used to replace either the yolk or a whole egg.

The best substitute will depend on the exact recipe and desired result. Applesauce and banana tend to impart added flavor, which can be a consideration as well.