Yes. It's called the optokinetic reflex if the eyes move, and recruitment if it's just the facial or eyelid muscles. If you experience these effects, you're playing video games too long. If these occur with minimal playing time, you should visit your doctor.
Dry eyes. Because people blink much less than normal while playing video games (or using a computer) eyes often become dry. The blinking and squeezing of the eyes is an attempt to moisten the eye with tears and is probably not a true tic. See if artificial tears help.