Would NOT accepting job offer w/Non-compete prevent me from collecting current unemployment benefits

The position I accepted did not include, nor was there any mention of a non-compete agreement until I was ready to fill out the employment papers. The previous employee in this position told me that he did not have to fill out a non-compete. I was not told about the non-compete contingency until after accepting the job offer, which is a requirement for employment. This contract would limit me from finding other work or accepting other offers, freelance, or the need to find a second job due to the very low starting pay of $15 per hour, should this position not work out for any reason. The non-compete would prevent me from utilizing a substantial job market, networking groups and businesses to draw from for freelance work or employment. This would be because the contract states that I wouldn't be allowed to contact clients or employees, for employment or otherwise, which happen to be current employment prospects. An example would be the Cleveland Clinic or Avery Dennison, both are current clients of the company, and I also have employment applications at or have done freelance work for. If Avery offers me a job at $25 per hour, I cannot accept it because they are both clients of and also a vendor of the company.

Bottom line,

Would refusing the offer, due to potential future employment restrictions, cause me to loose my current unemployment benefits that I have held since the beginning of May?