i used to fry an egg for my dogs every day - this was when i lived at home and treated my dogs like my kids lol - it made their coats slick as grease but that along with Pedigree dog chow gave them HORRIBLE gas!
you can feed your dog an egg every day but i cant remember exactly which part you remove, the white or yolk. one of them prevents the dog from absorbing something important if they eat too much of it.
other thing i do now is just throw a few eggs in the freezer(we have tons of bantam eggs from our chickens) so as a treat for the dogs they;ll get a frozen raw egg. keeps them busy, good for their coats, etc.