The details aren't too clear, but from what I read, I believe the facts to be thus: There was a problem and you reported it to your supervisor to find out what to do, only to discover they had no procedure in the event such a problem arose. Sometime later, your supervisor begins writing you up for things you believed you didn't do. Your supervisor now wants you to sign some paperwork reprimanding you for your supposed conduct.
Your question is should you quit and if you do, can you collect unemployment benefits?
To actually answer your question, you may only collect unemployment benefits if you became unemployed through no fault of your own. Quitting is considered a fault of your own, so it may not be the best solution to your situation. Some alternatives to your situation are going over your supervisor, assuming your boss has a boss. You could refuse to sign it and than allow them to fire you. This might sound like a bad idea, but you could than proceed to sue them for retaliatory discharge; termination in retaliation for calling your employer out on unprofessional, illegal or immoral behavior. Or you could skip all this tactic stuff and just seek damages for falsely accusing you of something you didn't do.
No matter what route you choose, I suggest you consult an employment law attorney. Best of luck.
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