Yup, over a week old, getting worse, and yellow and crusty = vet visit. It's probably a no big deal eye infection conjuctavitis, etc. type thing that will clear easily with proper meds, but it could turn into a HUGE sight threatening problem if you leave it.
I've never seen or heard of strangles only showing symptoms in the eyes. That almost always presents with a runny nose that progresses to swelling (eye discharge is probably not uncommon, but is more a secondary symptom). I highly doubt it's that; in fact would not even be considering it on my radar screen until a vet told me otherwise. It's still not a bad idea to limit contact with other horses until the vet sees him and wash your hands well after handling him (especially the "eye boogers") though.
I also would not just randomly put him on antibiotics. Antibiotics should only be used when you KNOW you are dealing with a bacterial infection (rather than fungal or viral infection or another problem genetically or environmentally caused). You also want to target the antibiotic type and length of treatment to the type of bacterial infection whenever possible. Just giving a course of antibiotics to "see what happens" when you THINK it might be bacterial is why we have such problems with resistant bacteria and hard to manage infections. The antibiotics won't work when you really NEED them, and that goes for YOU and all the rest of us to...not just the horse you used them on!
I understand vets prescribing broad spectrum antibiotics as a preventative for certain large wounds that are not only likely to get infected but likely to be hard to get un-infected once it's in there. This isn't the case, so call your vet and get the correct treatment for this the first time. That may be antibiotics, it may also just as equally be some sort of medicated eye drop.