Get them a hotel. Perhaps it will be comforting for them to know that this is not a huge airport and that it is also not a busy one.

If they arrive, take the hotel and then go up early in the morning, they do have a bit of refreshement and they will not be able to stay at the airport anyhow.

i would ask them why they do not want to leave - and i suspect it is reasons like being afraid. it would be different if it was over day, but just force them to their luck as with it "it is closed. you have to get out."

as 12 hours boil down to less hours with all the check in and out AND there is a hotel directly adjactant to it, this should ease the pain.

my suggestion: book a room at their airport with refundable price and tell them it is not refundable to take it. if they still insist, you can still cancel it.