Short version:
They make prescription eye drops out of the active ingredient in Chemiclean, so that isn’t your issue. I’d be slightly surprised if the nasty things on the SDS (not the active ingredient) both survive for 48 hours in our reef tanks without being broken down and get concentrated by skimmers. Perhaps RHF can chime in there.
Longer:
I’m making one assumption here, and that is that you dosed the Chemiclean to your tank some number of days ago, and have not touched it since. Thus the only path is from the tank water to the skimmer to you, not directly from the bottle to your eyes.
Through one of the local clubs I’m a member in I personally know a few people who have had palytoxin experiences that required medical attention. All of these people had been fragging their corals bare handed for years without issue before their palytoxin experience. Only one of them ended up with a crashed tank, or other major signs of disturbance beyond their medical issue. He was aware that something was off in his tank (it was crashing), had therefore taken care not to touch his face, and still needed a GP visit, optometrist visit, and at least two medications to stop the eye issues that ended up looking like a textbook palytoxin case compounded with an eye infection.
I’m good about eye protection, but poor about gloves. I’ve had years where I spend 500 hours with my hands in the reef tank, much of it handling coral. Perhaps once in that time I’ll get a short lived minor infection in a cut or cuticle, which some antibacterial bar soap resolves right away. Point being that doing something many times before without issue does not mean it wasn’t an issue this time.
Most of the nasty chemicals on the SDS I suspect are things that keep the erythromycin viable as a liquid while it sits on the shelf, or helps it dissolve in water better. I’ll need someone like RHF to chime in, but I’m guessing your tank processes those out in some manner, so they aren’t going to become super concentrated in your skimmate.
Consider the possible dose. I don’t know how much Chemiclean you dosed, but I’ll bet less than 100% of the active portion dosed ended up in the skimmer cup. Then dilute it by the extra volume of the skimmer, then take a tiny fraction of that and put it in a film on your hand, and a portion of that now ends up in your eyes. I don’t think it’s nearly enough to cause issues. I don’t know the concentration in the eye drops/creams that use it, but I’m guessing the total dose is more than the amount you could get via the path described above.
On the other hand, the odds of it being your lucky day to finally get a fish tank sourced eye infection seem comparatively possible. The feeling of an allergy attack is a known palytoxin symptom, and that is a chemical that does cause symptoms at the extremely low amounts possible by rubbing your eyes, breathing mist off of a skimmer, etc.
At best I think we could say maybe it was any of these three causes, but I don’t think we have a high degree of certainty that it was the Chemiclean in the skimmate.
All that said, arceye sucks, anything worse is just that, and I hope you are feeling better soon.