Some hens will lay eggs , they do not have to mate to produce them, of course they are infertile and you can throw them away. I would however inform the vet and show him/her the pictures. It takes a good deal of calcium to form the eggs and you want to make sure that her calcium is not being depleted. Did the egg have the hole in it when is was laid or did that occur because it dropped? It can be a dangerous situation if an egg is ruptured while still inside the bird. The egg could have had something to do with the consistency of the droppings.