Besides upper and lower eyelids, birds also have a third eyelid called a nictitating membrane which sweeps sideways across the eye. Birds can blink repeatedly with the nictitating membrane to clean the eye’s surface and spread moisture across the eye.
And yes they do draw this “rain shield” or a translucent membrane across to protect their eyes from rain.
It can also help prevent the eye from drying out when migrating long distances and hold woodpeckers eyes in place and protect them from flying debris as they peck a tree. Or in some aquatic birds like the Hooded Merganser the nictitating membrane is clear and acts like a pair of goggles to help the birds locate prey underwater.