Eligibility for unemployment compensation is determined by the laws in your state.  We cannot answer questions about state unemployment compensation here.  We answer questions about income tax and using TurboTax software to prepare tax returns.  You will need to check with the unemployment office in your state.

Since you haven't paid the UI Tax out of your paycheck, you can't collect Unemployment compensation.

As a sole proprietor your business is a disregarded entity. Therefore you do not issue yourself a W-2 or any other type of tax reporting document. If your state does not tax personal income that means that you do not pay Unemployment Insurance (or Unemployment Compensation Tax) for you - in the "traditional sense" as you may (or may not) pay it for your W-2 employee. Therefore different rules apply to you.

Since the UI or UC tax is a state thing and the laws on it can and do differ from state to state,you'll have to check with your state's unemployment agency to see if you meet requirements to qualify or not.