They need a good reason to fire you.
If you were doing your job there would be no reason for your employer to threaten to fire you.
It may not be the fact that you are a felon but the fact that you may not have told them and deliberately deceived them then yes they would have a right to let you go then. Employers do not need a "right" to fire for a particular reason. US law is clear - employers can fire you for good reason or no reason. The employer can ignore your felony or fire you for it or no particular reason, and need not explain themselves to you.
Employers can fire you for no reason at all.
Unfortunately, if you are an at-will employee (no union affiliation, no contract), your employer can terminate you for a good reason, a bad reason, or no reason at all.
Absent a contract between you and your employer or local law/company policy stating otherwise, an employer may fire you for any or no reason with or without notice.
Your employer can fire you with or without reason, unless it violates a statute.
An employer can fire an employee for any reason at all and need not explain to the former employee. Firing an employee for personal reasons that do not involve race, sex, age, religion, or disability is perfectly legal for employers of any size.
An employer can fire you for many reasons, but if you have been fired for something unfair, such as a health reason, you can go and see a lawyer and file a case for unfair dismissal.
I know in the state of Kentucky you can fire an employee for no reason at all within the first 30 days. After the 30 the employer needs a reason for the termination.
Short answer: Yes, your employer can fire you at any time without reason at all. BUT depending on your employment agreement it may not be a valid reason and depending on your location the employer may be responsible for paying in lieu of notice. To know for sure you need to call the government agency in your area and ask.
Unless a specific statute prohibits discrimination for reporting that particular crime, then yes. An employer can fire you for any reason or no reason, and needs to explain to no one.
An employer need not have good reason for firing you, or any reason at all.