An employer is not responsible for your unemployment benefits if you are an independent contractor. While employees are always eligible for unemployment benefits if they are laid off, an independent contractor will only be eligible if they pay separately into the state unemployment fund.
However, if your status as an independent contractor is questionable, filing for unemployment may be worth a try. While cases considering Uber drivers' employment status are still making their way through the courts, a Florida Uber driver recently filed for and was granted, unemployment benefits as an Uber employee.