I've been installing SyncBack on people's computers. I think it's a good program. It can back up to another folder, an external drive, FTP, whatever. I don't know whether it can behave exactly as you're looking for, though. The only scheduling it has is through the Windows scheduler. I just checked the options you have when creating a backup plan, and I didn't see any option to run at shutdown. That will probably be a tough thing to find, I would imagine.

I would suggest doing it the other way around. Find a program that will give you the option to shut down when finished.

But really, I don't think either option is very good. You shouldn't rely on manual backups. Get the software, and make it run on a schedule. Run it every day, or repeatedly throughout the day. I just checked SyncBack's options, and you don't even have to use the Windows Scheduler. You can make it run at preset intervals like every hour.


*edit*
If you want to check out SyncBack, click on the link above and scroll down to version 3.x. That's the free one.