Hi, I think I remember someone said that hair loss around eyes is a classic symptom of zinc deficiency. My dog gets this, so I'm about to get some zinc tablets tomorrow to see if they help. Though if your dog has gungy eyes, it could be something else like conjunctivitis and may need eye ointment/drops from the vets. My dog has hair loss, but his actual eyes are fine, not red or gungy.
Hilda