Bismihi ta'ala
No, eyes cannot get impure.
The eyelid can, because its external, and simply putting it under water would purify it.
And Allah knows best
Shaykh Dr Zaid Alsalami is an Iraqi born scholar, raised in Australia. He obtained a BA from Al-Mustafa University, Qom, and an MA from the Islamic College in London. He also obtained a PhD from ANU, Canberra. He has written and translated several Islamic texts and also prepared educational videos on Islamic rulings and practices.