solarz
Brigadier
It's called nictitating membrane. It is usually translucent and moves horizontally to cover one's eyes when needed (assuming you have one, of course).
I hate the idea that, evolutionarily "you lose it if you don't use it". That is simply wrong. Sadly, too many professionals still tend to think that way. Evolution only happens when there is pressure for it. For instance, hypothetically, 1% of human has wings. These people will slowly die off and at best stay as minority if nothing happens. If all of a sudden, a global flood occurs. The only way to stay alive is to fly away. Then these winged humans suddenly become advantageous. While most "normal" human drown, these winged human survive and have offsprings. Then before long, all human on Earth have wings. And you have evolution.
So the nictitating membrane must be disadvantageous for land animals or mammals. This should be why most mammals lost it. Not because it is useless. If it is useless, there would not be any pressure one way or another. Mammals would have kept it.
Any difference you see between human and any other animals should be caused by some functional advantages with how human ancestors behaved and lived back when human first evolved. These features may not be advantageous now because we don't have the same environment that our ancestors had back in the day.
So with the narrow eye lids mentioned by that doctor, that will never happen. There never be any evolutionary pressure to favor such physiology.
Evolution always always goes hand in hand with global disasters, the kind that will make a species extinct. Always! Without such disaster when we just go about our everyday life, evolution will never happen.
I think the "you lose it if you don't use it" idea is a simplification of a more complex idea, but it still makes sense because some traits, just by existing, have a detrimental effect to reproduction. Even if that cost is very small, over millions of years it will be bred out.
Take for example the human tail. There was clearly an evolutionary pressure for humans to not have a tail, however small it might be. There doesn't seem to be a physical reason for it. Having a tiny tail doesn't make you run slower, for example. However, there could have been social pressures: other humans didn't want to mate with a person with a tail, and thus the tail slowly died out.
Typically, traits that really don't matter to survival stay around, such as the different shades of hair color, or having double eyelids, etc.