Backgroud check only check the company you filled in the form.
Your previous compny should only give your start date and end date, they should NOT tell the reason why you left them.
How they know you were fired?
It is NOT necessarily for you to seclare.
If you are fired, you will not get UI.
If you get UI, you must not be fired.
Life are full ups and downs, please don't worry too much.
Wish you good luck and get a job soon!