Do birds get jealous? Can it make them sick or cause red eyes?

Birds can get jealous, and birds will each have their favorites they like or dislike, however liking someone doesn't cause redness of the eyes or other signs of illness.

The eyes look normal in the picture you sent, however, if the eyes seem very irritated, making sure there isn't a lot of dust or pollen in the area, and that any substrate in the cage isn't creating dust is a good first step. If you see any discharge or other symptoms of illness in addition to the eye redness, contacting the bird's local vet for an exam may be a good idea.