If you ever find yourself in this position, just leave the airport and get a motel room. The US doesn't even have exit immigration, so it's not like they were stuck on the wrong side with a used visa.
The OP's About page notes that they're currently unemployed and living off savings, so I'll cut them some slack, although I'm not entirely sure how that's compatible with international travel from New Zealand to the US.
I also find it incredible that the airline can just delay a flight by 24 hours and offer no compensation or accommodation whatsoever, since in most of the world this would absolutely not, ahem, fly.