The outside corner of my right eye is bright red. I'm taking Tylenol (acetaminophen) Cold for a sinus infection. Could that be the cause?

Not a cause. The fact that you're not feeling well may bring about associated viral mediated conjunctivitis, which may appear red, but usually goopy. The bright red spot may just be a bruise from rubbing your eyes. Some infections can cause bruising too. The Tylenol (acetaminophen) medication is not the cause though. Remember that if your eyes continue to be of concern, see an Eye MD to evaluate, esp., if no improvement.
Broken capillary. probably when you sneezed or coughed hard you broke a small blood vessel in the white of your eye. This is a benign finding and will resolve on it's own.it may take several days to do so. If pain accompanies the finding then see eye MD as pain does not accompany a broken capillary in the white of the eye.