Besides checking the "Disk Utility" - How did you connect the drive the Mac Mini?
Sometimes, some of the external drive might need "extra" juice to make it "run"
I found through "USB hub" would affect the issue, as well, my "home build" USB external drive has two USB connectors - one for data, one for extra power
I need to make sure both connected in order to get it going.
Some other market USB external drives do not need the extra juice - therefore, try other USB port to see if it runs.
Or - maybe the external drive is a "lemon" ??