Employee refused to sign a warning write-up, then quit, what is next?

Based on the few facts you have provided, I am not sure what you think you should do. Employees are "at will" in Illinois as a general rule. That means an employer can fire an employee for any reason or no reason at all as long as it is not a "bad" (illegal") reason (i.e.; unlawful discrimination, for whistle blowing, etc.). In this case, the employee quit rather than acknowledge sign the "write up". Absent other facts, the main issue may be whether the employee is entitled to unemployment. If the ex-employee files for unemployment, you should contest it on the basis that the employee quit and was not terminated by you.