I have tried using both a Vantec case and a Rosewill case. Both cases are externally powered. In both cases, if the USB 3 external drive is turned on when I boot up the Mac Mini, all I get is a grey screen with no Apple logo. If the external drive is turned off when I boot up, I get the Apple logo and it boots properly. Once it is booted I can turn on the external drive and it works fine. I have observed if the Mac Mini is simply being rebooted rather than powered on, having the external drive running does not cause the system to fail to boot.
Something to check. Power everything down. Turn on
your external drives then do a PRAM zap booting the MIni.
It may or may not help.
If it doesn't, then check the system log starting from the time
you start up the Mini and see what the last entry was. Wait a fair
amount of time so a gap in time in the log can be seen when you turn
the drive off. Could determine if it is some hardware issue, as BDAqua
was suggesting, or a driver issue.