There are really two potential answers to your question.  Legally, in the State of Texas, a person can record a conversation with another person if at least one party to the conversation is consenting-- consent from all parties is not required.  This means that if your husband is a party, he can legally record the conversation he has with anyone-- including his boss or supervisor.  The second answer, however, applies to employee handbooks.  Even though the State of Texas may be okay with the recording, many employers write rules into their handbooks that prohibit the recording of conversations-- so that part of your question will turn on your husband's employer's policy's.