Eggs are great for dogs. Full of protein, and a relatively cost-efficient source of it too, they also contain all the amino acids required by dogs for good health and are high in iron, selenium and Vitamins A, B2 and B12.
There are some misconceptions about feeding dogs eggs but actually the risk is really low. Firstly, the risk of salmonella (approx <1 in 20,000) which can be significantly reduced by feeding fresh organic eggs from happy, free range hens, and storing in a cool, dry, place.
The second risk is the biotin or vitamin B7 inhibitor called avitin contained in egg whites but in fact you’d need to feed dogs many eggs per day to cause a biotin deficiency and you can also counteract any possible effect by feeding the egg shells too, which are rich in biotin as well as calcium. Incidentally if your dog won't eat the shell you can bake them and grind into a powder to sprinkle on food or buy powdered eggshells (which I buy for Gracie at Meiko).
It is not recommended to rely on eggs as the sole source of protein in your dog's diet but as a general rule a couple of raw eggs per week along with a balanced diet (see here for more details) will provide them with excellent nutrition.
It's also ok to feed cooked eggs if you prefer (boiled preferably and definitely not fried for health reasons). Eggs can help upset tummies but for maximum nutritional benefit, raw eggs are preferable as cooking destroys vitamins, minerals and amino acids. But even cooked eggs will provide a good nutritional boost for your dog.
Like anything, if you notice your pet having any digestive issues when you introduce eggs to their diet, stop the eggs and if necessary consult your vet. For more reading on adding eggs to your dog's diet click here and get cracking (sorry...)