Well, it 'can' happen if bodily fluid gets into the eyes. Symptoms are similar to that of pink eye. And if it happens, a person needs antibiotics. Is this something you are concerned about for yourself currently?
People who have oral gonorrhea or chlamydia aren't infectious by sneezing - it's not airborne. It's only infectious if they give someone oral sex.
Getting one in the eye usually happens via childbirth or someone autoinoculating - touching their symptoms genitally, then touching their eyes. It doesn't transmit from eye to eye like you think. If you touch someone's runny eye and then immediately touch your own, it could, I suppose, but how often does that happen? Gonorrheal and chlamydial eye infections are rare.