The reason that some people recommend you only feed the egg yolks, or that you should only feed boiled eggs (and never raw) is because egg whites contain a protein called avidin.
Avidin binds to biotin, and if you were to feed your dog only raw egg whites, it would create a biotin deficiency. Fortunately for us, egg yolks continue plenty of biotin, so if we feed whole raw eggs, there really isn't anything to worry about.
You can feed eggs raw and whole, including the egg shells, if your dog will eat them. Abby didn't used to like the shells, but Ronja doesn't mind them. It comes down to personal preference.

I've never had any tummy issues due to shells, personally.