What is my husband’s obligation regarding an overpayment?

Signing the agreement was the wrong thing to do. While the law does say that if there is a legitimate or true overpayment due to a mistake or error, that you have to repay the money--a mistake does not let you keep money to which you are not otherwise entitled--the party demanding repayment would have to prove it if you refused to repay it. So your husband could have simply refused to repay if they did not provide the amount of the overpayment and prove it to his satisfaction, in which case they would have had to sue him for the money and prove their entitlement to it in court. But by signing the agreement, your husband contractually obligated himself to repay an amount which he's not even sure of, and without it having been proven that he was overpaid. That will count against him, since the law presumes that if you sign an agreement, that you did in fact agree to be bound by it. It will be hard to escape paying $250/week unless you can show that there was no overpayment or it has been repaid.
That said, what your husband can do is:
1) Compare his hours worked and his pay rate vs. what he was paid--he should have this information. (If he did not keep records, then for future reference, *always* keep your own records of your time.) If he doesn't have it exactly, use a reasonable guess, like if he worked 35 hours per week on average for 20 weeks, he worked 700 hours; if he was paid $20.00 per hour, he should have received $14,000. If he received $14,000 in this example, he was NOT overpaid. But if based on pay stubs, direct deposits, etc. he received, say, $25,000 during this time, then he knows he was overpaid--he received around $11,000 more than he should have.
2) If he was overpaid, then having signed the agreement, the best thing to do is to honor it UNTIL his calculations show that he repaid the money (he obviously needs to track repayment), then tell them to stop it: at that point, he needs to not repay anymore.
3) If by his calculations he did not overpay, then refuse to let them have any money starting right now; let them sue him for breach of the agreement if necessary, and again, they will have to prove their case in court.