Were they not required to sign a list of people who have voted in order to be given a ballot?
Probably not. I know in my district you don't, and it's probably the same in New Hampshire. Given the circumstances, that you are at a polling place, and the context of the conversation, I would think once accept the ballot, that would indicate confirmation.
Actually, once you indicate a name and address to the clerk, which they then draw a line through on their list, that could reasonably be assumed as confirmation.
Personally, I think they handled this very badly all around. About the only thing they really accomplished is video taping themselves engaging in voter fraud.