It could mean a lot of things. To be sure you will need a vet to examine and determine what is happening. Do not let this go, it could be serious. I am speaking from experience here. I could list a number of things, but that would be pointless. The horse needs to be checked by a vet.
Jemma and Kiara had red runny eyes several times after we had really strong wind and there was all this dust flying around. I made very strong tea and just cleaned their eyes with the cotton ball soaked in warm tea 2-3 times/day. It seems to work for me. I also did the same with my cats and dogs when they had similar problems.
VanillaBean, is your horse's eyes still runny? If so, you might want to take a cotton swab with warm water and clean the area where it's been running on it's face. This can be very irritating to their skin. Make him/her a little more comfy till you see the vet.
Keep us posted on the results and good luck. Hopefully it will be simple, if not, better to be safe.
No the vet didnt show me, i wasnt here when they flushed it, but Sheena had to be drugged so i'm thinking it is not something that i can do because Sheena is really well behaved, so if the vet drugged her, it was necessary.