I would claim that your refusal to sign a non compete is not the type of misconduct that would justify nonpayment of unemployment. For some bloggings on this topic, check out my employee rights blog www.takethisjobnshoveitblog.com or my website www.behrenlaw.com.
You can forfeit unemployment by voluntary quitting or being fired for misconduct. You should check with an employment attorney in your state to see if this action will forfeit your rights to unemployment.
THESE COMMENTS ARE NOT LEGAL ADVICE. They are provided for informational purposes only. Actual legal advice can only be provided after consultation by an attorney licensed in your jurisdiction. Answering this question does not create an attorney-client relationship or otherwise require further consultation.
Normally a voluntary termination excludes you from unemployment compensation. There are exceptions if it could be considered constructive discharge. You have a valid argument that you were to be terminated for refusing to sign the non-compete agreement. You should contact an unemployment attorney for representation.