Everyone's eyes twitch when if they're drunk enough. When cop/doc/whatever tells you to look at their finger as they move it side to side, they're not watching to see if you can follow it - they're watching your eyes twitch.
It happens because your inner ear is part of a reflex that your head uses to keep your gaze steady when you move your head. When you drink alcohol, it gets into your inner ear. Because its less dense than water, your body interprets this as spinning (which is why you get the spins if you drink too fast). Before you feel actual vertigo, your eyes will twitch as your inner ear is telling your eyes that you're spinning, but your brain is telling your eyes that you're not.
Or, you probably have AIDS