Mid '11 Mac Mini won't shutdown cleanly. OS X El Capitan 10.11.6
Firstly, you should update to OS X 10.11.6. It will solve many problems. Download 10.11.6 Update.

Secondly, reset SMC. Here is a link how to do it: Resetting System Management Controller.

If it doesn't help, try clean install: How to Clean Install El Capitan.

ADDITION:

If you can't do clean install using your own mac, make a bootable USB on another mac: Click here.

Also, go to Utilities and run Console.app for inspecting what's wrong with your OS: Click here.

It might be a problem with a hardware, so try single-user mode pressing cmd-s after startup 'chime' sound or verbose mode pressing cmd-v the same way. To read about it Click here. 

And read an important post about 7 Mac Startup Options: Click here.