Not exactly a hack for getting shampoo and conditioner out of your eyes, but try avoiding getting them there in the first place. Tilt your head back as you shampoo or condition your hair and the shampoo and conditioner will tend to drip backwards instead of down your face. For the duration in which the shampoo or conditioner will be resting on your head, try to avoid tilting your head forward. Also, if you feel as if the stuff is starting to get into your eyes, close them, then wet your hands and brush the stuff away.
On the other hand, let's say you manage to get shampoo or conditioner - or even soap for that matter - in your eyes. First, get the water running and cup your hands in the stream. Take this water and gently press it into your face (don't jam it into your face), wiping slightly. I've found that this always works for me.