We have a consumer account in which the authorized signer cashed a check and it was returned to the bank as account closed. The authorized signer will not come in and give us the money back. Do we have Right of Setoff in this situation on the account in which they are authorized signer?

Most likely not. An authorized signer is not the same as a joint owner. While they have the authority to transact on an account, that doesn't give them the right to use that account as recourse to conduct their own personal transactions.

To be sure you would need legal counsel to review the document that the account owner executed granting the authority to the authroized signer to see if there is any language about the account owner accepting legal liability for personal transactions performed by the signer that are not related to the account.

This is the reason that most banks will not cash checks payable to an authorized signer only on the strength of that tenuous relationship.