You can certainly make the substitution, but you’ll have a totally different result. Egg whites are a protein and help bind the pumpkin--i.e. make it more solid.

I don't think so, the egg whites would most likely be a binding agent much like the whole eggs are in a regular pumpkin pie recipe or if they are whipped they add volume applesauce would do neither. Applesauce is sometimes used instead of some or all of the fat in recipes so I don't think subbing applesauce for egg whites would work but I haven't seen the recipe so don't know the specifics.