Or would that eye be the most well-defended from external debris and whatnot in the entire world? Cats seem to get by just fine without eyelids and a membrane but no eyelashes...likewise humans with lashes and lids but no membranes...
Cats actually have eye lashes, two eye lids and a nictitating membrane. They also have furry eye lids. Unless the cat suffers from trichiasis, there won't be a problem.
Ostriches as well:
Humans have what's left of a nictitating membrane, the plica: