Dogs don't really have dandruff like people do. It could be dry skin and you could need to do something as simple as adding humidity to the house. Or, more common dogs have skin conditions as the result of allergies. Have you changed diet, or exposed the dog to new conditions around the time the people started? A quick trip to the vet might be in order. Also, how often are you shampooing? Some conditions require more or less bathing. And there are special shampoos specifically for pups or for allergy-prone dogs. Try these things and see if any of them help.
As for egg white, it might work, but I wouldn't feed raw egg to a puppy for fear of salmonella.