Yeah, it is possible that coughing may cause red eye or pink eye, especially pink eye. Redness in eyes does not mean you may or already get pink eye, or conjunctivitis. We have known that coughing is commonly due to cold. While cold is most possible because of bacteria and virtues. And it is known that pink eye or conjunctivitis is most possible to bacteria and virtues enter eyes. That is, bacteria and virtues can cause eye infections, pink eye is one of them. By the way, pink eye is not serious and you will be fine without any treatment options in 4-7 days. And don't rub your eyes with your dirty hands, even you think you have already wash them up.