The fly mask is a great idea. I had a filly who had runny eyes from the day she was born. I never had her tested but I assumed it was allergy's. I think as long as the discharge is clear or whitish I wouldn't worry about it. I always kept a hanky on hand to wiper her eyes.
Vida's eyes will run when she is on hay in the winter. She clears up in the summer time when she is back on pasture, but they all like to bury their heads in the hay bale so its normal for them to get runny eyes and as far as I know theres nothing you can do about it.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France