'Green eye' in horses is the same thing as red eye in people, caused by the light reflecting back directly towards the lens, and resulting from having the flash far too straight on to the subject.
It would be easy enough to correct PP, simply do a very rough selection (obviously don't include any grass) and then in PS, go to image>adjustments>selective color>greens and adjust the sliders as necessary.
Or just light it properly in the first place.