Eggs have a number of benefits for dogs. They are loaded full of protein and healthy Omega Fatty acids that will help encourage a slick, shiny coat and repair tissue. Why? Because Omegas and protein, among many things, encourage healthy cell reproduction. And skin, hair and tissue are ultimately cells! The fatty acids also support natural oils secreted from pores to protect hair and skin.
It's a good idea to consider that typical grocery store white eggs are from chickens fed poor quality feeds and by-products left from the manufacturing of other substances including alternative fuels, cheap beer, and high fructose corn syrup just to name a few. Whenever possible seek out organic, free range, and farm fresh eggs. Healthier chickens produce eggs higher in healthy fats as well as vitamins and minerals.
Shells are generally considered a great source of calcium and protein. Crushing or grinding them up can help make them easier to digest.
The choice to cook or serve eggs raw is up to the individual owner. There have been no reports or death occurrences in dogs as a result of contracting deadly bacteria or diseases from consuming raw eggs that I can find. Salmonella is very rare in eggs whether humans or dogs are consuming them. There is speculation that avidin which is a protein found in very small levels in egg whites can effect the function of biotin (B7 vitmain) in the body. However, further careful research shows that there is enough biotin in the egg yolk to eliminate the concern for excess avidin.
If you prefer to feed your dog eggs that have been cooked you can boil or scramble them. Some fatty acids and fat soluble vitamins will be lost to the cooking process. But the overall impact of the eggs will still be beneficial.
For a large dog, such as a Rottweiler, an egg per day is a great addition to any diet. More can be given to pregnant or lactating bitches; less for young puppies or smaller breeds. Eggs are a fantastic, well rounded, "mother nature's perfect food" addition to first foods for a litter of puppies. For puppies just learning to eat solid foods scrambled eggs are a great place to start.