Cooked eggs — yes, happily. They're a good source of protein, amino acids, and several vitamins, and most dogs eat them readily. Scrambled, hard-boiled, poached — the cooking method doesn't matter much as long as you're not adding butter, oil, salt, or anything else.
Raw eggs are where it gets a bit less clear-cut.
Why cooked beats raw
Two issues with raw eggs specifically. First, raw egg whites contain a protein called avidin that binds to biotin (vitamin B7) and prevents its absorption. A dog that eats raw eggs occasionally probably won't develop a biotin deficiency, but regular raw egg feeding over time could cause one. Cooked egg whites don't have this problem — heat denatures avidin and makes the egg nutritionally straightforward.
Second, raw eggs carry a salmonella risk. Dogs are generally more resistant to salmonella than humans, but it's not zero risk, and handling raw eggs and then feeding them to a dog that then licks things in your house creates some indirect exposure for the humans too.
Cooked eggs are just as nutritious and eliminate both issues. There's no practical reason to feed raw.
How to prepare them
Plain is best. Scrambled with no added fat, hard-boiled, or poached all work well. No salt, no pepper, no seasoning. Definitely no onion or garlic — both are toxic to dogs and sometimes sneak into scrambled eggs as a habit.
The egg by itself is what you want, without the way humans typically cook eggs for themselves.
How much
Eggs are relatively high in cholesterol and calories, so this isn't an every-single-day food. A medium or large dog can have one egg a few times a week without issue. For smaller dogs, half an egg is plenty per serving.
As a treat or a food topper on their regular meal, eggs work well. They go nicely alongside plain cooked rice for dogs recovering from an upset stomach — that combination is a classic bland diet option vets often recommend.