Q. Can eye redness be caused by leash pressure on the dog's neck during a training class?
A. Absolutely. Pressure on the neck restricts major blood vessels that lead to the eyes (and other areas) which can lead to pressure on the eyes causing red eyes.
As long as the eyes aren't runny or have a milky/yellowish colored discharge, that would indicate infection and a trip to vet.
Try switching to a front attach harness to help with training. That will take all the pressure off the neck (and those arteries and thyroid gland too) while still helping you with training.