The eyes are a great way for the virus to enter the body.
Face screens are highly recommended.
I doubt this is true.
It is certainly a concern for medical personnel who might be in close contact with CV positive patients. China learned that from SARS back in 2003.
Your nose and mouth are actively vacuuming in air and its attendant particles. Eyes are passive in this regard.
Touching your eyes with unwashed hands poses some level of threat, but you're already talking about a few degrees of remove. You need a virus contaminated surface, touched by hands and then hands to eyes, all while the virus stays alive.
I'm sure it happens and that's why they talk about washing your hands and not touching your face. But this is a respiratory virus and respiration is the major method of transmission. Hence the need for face masks.
In early March my one neighbor wore protective goggles along with her mask and gloves to go to the food store. Some scattered folks wore eye protection. I might have considered it if Shanghai had a major outbreak. I did frequently put on my light winter gloves when I entered a store, as they were in my winter coat pocket anyway.
Really, the more precautions the better. And the higher the compliance rate, the better. But you also want to match your behavior to the threat environment.
I'd say Trump's USA failed on all three of those metrics.