What do I do if I got a little peroxide in my eye?

Flush it out. Peroxide in commercial concentrations will do no harm to your eye if you got a splash or two into your eye. You could flush it out if uncomfortable but do not worry. See an ophthalmologist if you have a change in the vision, have pain or redness - but these are very, very unlikely in this circumstance.
Rinse immediately. Peroxide is very irritating. If you get any in your eye, rinse your eye immediately. Should you still experience any significant pain, see a doctor.