I agree. If your husband's former employer has not paid him for the last day he worked, that is a legitimate complaint that can be dealt with by wage and hour. Vacation pay? No, they don't usually have to give that to you, though I'd ask Wage and Hour about that too and discuss it with them regarding his unique situation.
If he was let go due to a clear cut attendance policy and he was aware of how many days, points etc. he had, in other words if he had received warnings about his attendance before his termination, his chances for unemployment insurance aren't great unless he had a medical excuse for the very last absence, the one which triggered his termination. But as has been said, it costs nothing to file and get a decision about his eligibility.
Submitting resumes does not usually trigger a reference check, so those no callbacks from resumes probably is not related to his prior job giving him a bad reference. The interviews? Possibly, though the employer is always allowed to tell the truth without repercussions. If he was fired for missing too many days, that's what really happened. A lot depends on how your husband is fielding questions about his past employment in interviews. If he responds to a question about "why did you leave your last position?" with a litany of how "they done me wrong" or sounds the least bit culpable or resentful about his previous employer, that could be causing the interviewer to pick up a bad vibe, which they are totally allowed to do.
"Why did you leave your last job?" might have many answers which would not set off the alarms, without hurting the truth too badly. He may want to give this some thought. Especially if he was out due to health reasons or because of a family member's health issues which have now been resolved he may want to try to deal proactively with this possible poor reference. In any case, he'll want to concentrate on the positive aspects of his past job, the good training he received there, the opportunities to learn and grow, the years of faithful and appreciated service....not just the final outcome and why he was terminated.