I got to the part where they tell you to never run superhot.exe again and I quit the game for a bit, but when I start the game up again it starts at the beginning of that mission again. It's happened 5 times now. Is it supposed to save the game at the end of the level? Is there a way to force it to save before I quit?

I stopped at the same point first time playing. Coming back I had to redo that part (click to walk upstairs and see they're watching you). Unfortunately, this is only a couple levels from the 'end' of the game, so it was better to go ahead and finish. Probably why the last checkpoint is just before this.