The answer is: depends. If you have pre-existing conditions, or slow reaction times, you may need eye protection.
I've taken so many balls to the face (including direct smashes) that you'd expect me to be hit in the eye, but it hasn't happened once because I have existing reflexes that help to protect my eyes. Then again, I spent years being punched in the face, so that's probably where that comes from.
Also, the risk decreases as you progress in levels, because you learn better positioning and better players are very unlikely to send a ball at your head (since it'll be out if you duck).
So overall, I think that lacking a pre-existing condition, eye protection is probably not quite so important in padel in the long term.
Naw, don’t listen to this guy. Poor reflexes aren’t the only way a ball can fly into your eye