This advice was given to me when I suspected a chicken of having gapeworms - To check for certain for gapeworm, my vet showed me this: Put her in your lap, take a Q-tip and gently open her beak, holding it open with one hand. Take the Q-tip and gently put it down her throad, TWIRLING AS YOU GO DOWN - you don't have to go far. Twirl as you bring it back up, and if she's got gapeworm, you'll see thin, red strings on the q-tip.
I am guessing that it is probably something respiratory since she also has the eye thing going on. I don't think gape worms would cause that.