Indeed cats can catch conjunctivitis, or they may develop conjunctivitis due to stress. It can be easily treated with some eye cream which is only available on precription from a veterinary clinic.
Usually if a cat has conjunctivitis its eye will appear to be semi shut initially, there may also be a little discharge from the eye. As the condition progresses, the eye may close completely.
The eye cream will soon get rid of the condition, but often the cat will lick the discharge away from the affected eye and therefore it will appear a day or so later in the unaffected eye. This is why proper treatment should be obtained, otherwise re-infection is common.
Sometimes the cats will persistently wipe away the eye cream once it has been put on the eye. To stop this your vet can prescribe an 'Elizabethean Collar' which is a large collar which means the cat cannot wipe its eyes. They also hate wearing them !