I had a similar situation ten years ago with a stray that had an eye infection.
I knew somebody that gave me some drops, IIRC it was amoxycillin, and the infection cleared right up for her. I bet I might even be able to find that bottle to see the name, if you're interested.
At least take him to a vet or shelter to have him scanned. If he has a microchip, they can contact his owner.
If You are able to take him to a Vet and keep him I disagree with putting him down.
It could be a classic case of cat flu- found in a lot of strays. I myself had a kitten I found not long ago put on medication for it and anti biotics not long ago and he is now a little ray of sunshine
He needs to see a vet regardless
could you please find it in your heart to take this poor kitty to a Vet and have his eye looked after.....is there any possible way you could consider keeping him as a pet? you seem to have the compassion to care for him and he would give you so much love in return....otherwise hon, if he is suffering and no one wants the poor boy than IMO the only humane thing to do is have him put down, he cannot be left to freeze and suffer....-(