Welcome to JustAnswer. Pink eye (conjunctivitis) can be caused by viruses, bacteria, fungi, irritants, allergies, chemicals, etc. Without knowing what is causing it, there is no way to tell if amoxicillin would help or not. Generally eye drops rather than oral medications are prescribed for conjunctivitis once it has been established that it is a bacterial infection.
Yes, it is a sad fact that a lot of people don't have health coverage. However, there is no way to tell if it is viral or bacterial without an exam, and amoxicillin will do nothing at all for anything but bacterial infections. It's also true that most upper respiratory infections are also viral. I would not waste the amoxicillin, and you may find artificial tears give you more comfort than allergy drops (unless it is an allergy). You can check out free clinics in your area. A GP or IM doctor, NP or PA can prescribe any medication that you need. Most doctors also will work with you on monthly payments and/or take credit cards. So will most ophthalmologists.