An employee we fired last year came in to our office and wanted to reapply again, do we have to let him reapply?

By reapply if you mean fill out an application, then the answer is yes.   By law, you must allow anyone that comes to you to apply for a job.   Even if you don't have an open position, they are allowed to apply.  You don't have to hire them and you don't have to interview them, but you must allow them to fill out an application.   It is also best to keep completed applications on file for at least a year. 

Remember, an official application form is a legal document.  A resume is not.