Ok, it is quite of you to make handstands, which requires a lot of strength and fit body. Anyway, handstands are not a very healthy habit, and it would have some negative impacts on your health and side effects could emerge. Anyway, try to make less of it and take care of your body. You should know that when your body is upside down, something would go wrong.
Yes, handstands will be bad for your eyes because of the rushed blood circulation at your whole body. Your eyes may easily get red when you handstand. You should not do it for a long time. If you feel not comfortable at the eyes because of the turned blood circulation, you should have the good rest for the eyes and use the warm compress to release the symptom and make your eyes feel comfortable.