I am experiencing something similar just recently, after having practiced the headstand for extended holding times for many years. A response I received is that the tissues around the eyes are very thin and sensitive, and if the blood is unclean, and then you are standing on your head and increasing blood flow to the area, that toxins will easily escape through these tissues...hence the reddish, "burst blood vessel" experience.
The advice you have received here is good. I would ask whether you have attempted any cleansing? What is your diet like?
I am undergoing some strong hormonal shifting recently and am continuing to cleanse and finding some amelioration of the "eye thing".
Just wanted to offer my words, since it seems we are experiencing the same thing.