I actually saw this episode on TV one day while home sick so it's been around awhile. Didn't bother to see if it was going to be a repost from years ago because it still applies to today.
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No, he has a moral obligation to share rent expenses with her if he made a verbal agreement regardless of whether he signed the lease. Beyond that, the point Judge Judy was making was that he took taxpayer dollars ostensibly to pay rent so he had a responsibility to either pay that money toward his rent or return it. The stipend funds were not provided for him to go get MMJ or Mad Dog. She dismissed the case because the woman hadn't paid the rent after the first month either so had no legal claim against him (the landlord on the other hand has legitimate grounds for a claim against her and possibly him).