I never did this, but I have acrazy idea.
Due to inexperience I need people to back me up or disupte this, but it makes a bit of sense: it takes a good deal of light to catch a reflection in someone's eyes, as well as a good macro lens (um...got one?) and really, really sharp focus.
So...
I'm thinking - don't use big aperture - go smaller (f/8 or higher, at the distance required to catch an eyeball) in order to minimize depth of field.
No flash? No problem! But you need a few light sources and a lot of practice shots: the eye itself will need to be lit, and you will need to be REALLY REALLY lit. So: you stand with bright mid-morning sunlight directly on your face, and have a reflector (no reflector? Use a sheet of foamcore or paint a big-ass piece of cardboard Navajo White or something cheap) reflecting light on her face, specifically her eyes. Control the lighting only by changing shutter speed and/or moving the reflector to increase or decrease the light on her eye, which will increase/decrease the comparative exposure of your reflection.
I'm thinking a triangle - you facing sun, her ahead and to the left or right a bit, and someone reflecting the sun on her eyes at an angle. 45 degree angle? 70 degree angle? I have no idea. I think the reflector should be close to you, but not behind you.
So: You'll be brightly lit (sun), her eye will be well-lit-but-not-as-lit-as-you (reflector), and...
I'm already out of my depth, but it sounds like a great workshop assignment to me! If you don't have tripods and stands and clamps, you'd better have a few helpers around. Hey: offer to buy classmates a beer (if legal, sonny!) for holding reflectors and stuff.
So. There. Well. Maybe I'm stoned.
I dunno - I have ABSOLUTELY NO experience with this. Comment is invited.