Might dogs spread eye infections to humans?

Unlikely: Most organisms are species specific and so a dog with an infected eye will not have a species that will transfer to humans. There are a few cross overs like rabies that can affect any mammal.

Yes but rare: These are called zoonoses and can cause conjunctivitis and inlamed tear ducts. They are easily treated with common antibiotic drops and ointments. Dogs are considered clean compared to cats.