There is no wrong way to feed eggs, really. Raw is more nutritious as nothing has been damaged from cooking. Cooking the eggs reduces the digestibility of protein from 90% to 50%. However, raw egg whites can be a little harder to digest than raw egg yolks. As rhondas mentioned, egg white contain an anti-nutrient called avidin. About 70% of this glycoprotein, avidin, is destroyed when the egg white is cooked - which is a good thing. Most of the good stuff is in the yolk. So the best way to feed eggs would be mostly raw egg yolks with a lesser amount of cooked egg whites. Shells? You can feed them if your dogs need more calcium in their diet. If you are feeding egg shells, you should feed less bone.
Here is an excellent article on egg whites vs egg yolks: The Incredible, Edible Egg Yolk
Notice that egg yolks contain 90-100% RDA (for humans) of many nutrients.
The only important things that egg white have more of is magnesium and niacin. Also, egg whites do not have more protein than the yolk ounce per ounce. White appear to have more protein on the chart because there is more volume of white in an egg than yolk. If you take an ounce of yolk and an ounce of white, the yolk has more protein. There are about 55-60 calories in a large egg yolk and about 15-20 calories in a large egg white.
So in my opinion, an egg snack consisting of 3 raw yolks to 1 cooked white would be quite nutritious! Double that amount for a full meal. Personally, I don't usually want to bother with cooking the whites so I just feed the raw yolks and toss the shells.