Before the possible fix, you should pinpoint the source of the problem. That is; you need to see the text messages behind the screen with the Linux Mint logo and a progress bar. There are a few ways to do this:
1. While shutting down, press Alt+F3 (F4, F5, F6, F7, etc.) to find the text screen whence the messages flow.
2. While starting the computer, at the Grub menu, press F2 to edit the lines and remove the words "quite splash" from the kernel boot parameters. This will temporarily disable the a/m shutdown screen when you use the panel later on to shutdown the computer.
3. You can have a look at the system logs (available under Mint Menu >> Administration tools) for an indication of the stall.