Hello, usually, if the corner of your eyes hurt when you blink your eyes, it must be the reason that you get some problems about your eyes, such as infection or injury. But since the organs of human body are related, sometimes sinusitis and some other problems can also cause the hurt of your eyes corner. Finally, it also can be the symptom of local eye keratitis.
There must be some problems for your eyes. You may use the eyes to see the computers for a long time with no rest and cause the eyes to be dry. That is why your eyes feel hurt when you blink your eyes. You may go to see the doctor and have a full check on the eyes.