Q: After installing El Capitan my Mac Mini won't boot.

Thanks or the info.  I accidentally started up Internet Recovery holding D down and so I'm seeing if that gets anywhere.  Safe boot appears to be broken as well.  I hold down shift while powering up and it doesn't appear any different.  The progress bar goes about to 3/4 and then the Mac turns itself off again.  It doesn't lock up.  It just keeps turning off before completing the boot.

I'll head back and try the NVRAM clear options if this doesn't get anywhere.

Well, this "it just works" thing is pretty bogus.  "Starting Internet Recovery"  --hours later and it says it's starting and may take some time.  However it gives no useful information on what is going on, no status bars, and the DSL modem Internet light is rarely flashing.  So it doesn't seem to be doing anything with Internet traffic.  I'm thinking Internet recovery is broken too.  Oh, how a simple progress bar or status indicator would be so welcome.

Many things Apple just don't work well.  Like this discussion board.  I had to go use my Mac because it won't work on the iPhone.  Java links don't work even though Java is enabled in iOS Safari.

The good news is that I was able to recover (upgrade to El Capitan) my Mac system without using Time Machine or losing data.  Apparently it only corrupted the OS install.  In my case it was the lack of information that was holding me back.  Apple is good about hiding what's going on but that can be detrimental when things don't work.  They just kind of leave you hanging and don't program in any helpful info or logical results like timing out when things aren't working.

Since the OS was corrupt I tried a recovery over the Internet.  In my case, it sat for many hours saying that it would take a very long time and spun an icon like it was doing something.  It turns out it was doing nothing.  Since it cannot load my settings it was trying to get a DHCP IP address assigned.  I have no DHCP server on my inside network and for whatever reason it thought if it tried for eternity it might eventually get an address.  Had it timed-out or shown a status that would have clicked immediately.

I moved the Mac to an outside network and my DSL router assigned a DHCP IP address.  I then did Internet Recovery since I have no local media with El Capitan on it.  I don't recall what it was showing but it was much more apparent it was actually working.  It re-installed OSX El Capitan.  I chose the choice that sounded most likely to be install over the existing one or repair the existing one.

Once it was setup I ran into another problem, with iCloud.  It kept claiming that there was a problem with the Apple server... well, after more troubleshooting it turns out it had no Internet connectivity.  You'd think that would be an easy test to see if you can route out.  A lot of my appliances make such a test and provide that status. --Anyway, the issue was that my data was not lost and so after part of the setup it regained my data and was using my static IP address again.  That won't work on the outside network.  I moved the Mac back to my inside network and was able to finish the setup... well, almost.  Later that night finishing everything it was giving me weird database errors in the iCloud login message.  Not user friendly.  I noticed my iPad wouldn't work on the App store and found the Apple Status webpage which showed that the Apple Store, iCloud, and some other services were down.  Sigh.

Right now I am back with El Capitan and my data is all there.  I'm not noticing any issues with the OS.

I'd recommend trying the Internet Recovery.  I think I used command-option-r while booting.  Here is the list of boot option keys you can hold down while it starts up:  https://support.apple.com/en-us/HT201255

Hope that helps --one of those options should get you a running OS which hopefully will install correctly over anything that may have been corrupted.