Can the State of Minnesota or the IRS take money from a bank account that is not a joint account to pay for a spouse's back taxes if the one who has the account does not owe taxes?

Minnesota is not a community property state, so the answer is generally no.

However, if the funds are in the account as a result of a fraudulent conveyance to evade payment of taxes, action could be taken to recover them.

Errors do happen, of course. As long as you are married to someone who does not pay their taxes, you do have to worry.