If you are paid with a check from a bank that you owe money(from one of their customers), will they keep it?
No, and you don't have to go to that bank to cash a check. The bank's name on a check does not mean it has to be cashed at that bank.
They will not keep the money, unless it is something like an overdraft. You can cash the check and your money will not be affected. However, you will need to talk to the customer so that you can agree on the mode of payment.