I'm not a vet, but I have seen that type of eye swelling on tv vet shows before and know that in a chicken, the pus tends to solidify. Watched a vet on tv basically try to manipulate the lid open and pull the pus plugs out with tweezers.
If it was me and you can do what you want, I would prep a small area(clean it good) and take a sharp sanitized knife and make a small cut and see if I could squeeze or pop the pus out that way. I'd wrap that rooster up real snug in a towel before trying. I'd have some type of eye antibacterial ointment to put on the wound after pulling out the pus and hope it closes up on it's own. That would be the problem depending on the cut. Could need a stitch or two. I would try to get it out through the lid opening first though. I have my doubts. It looks huge!
I don't know it could be a blocked tear duct? But every time I saw one of those on tv, it was solid pus. Some type of eye infection or maybe sinus infection in the chicken I guess.