eggs are excellent for dogs.
how many to give really depends on what else you feed the dog. a large egg has about 75 calories and an extra large about 85, and most of them are fat calories.
if you feed kibble, i'd say you can include an egg several times a week or even on a daily basis.
the fat in eggs is mostly saturated, which does make for a shiny coat, but omega 3 fatty acids affect coat, skin and many other systems in the body, so you have to make a decision on what kind of fat calories you want to include. personally i'd rather go for a good quality fish or salmon oil.