Some drive enclosures have power switches, but they tend to be either on or off. The Mac Mini power button is sprung so it only stays on for brief time. You will need some kind of extra logic in order to make it work.
If you are on a budget, OWC do Newertech USB2 to ATA/SATA adaptor for $25 or so. Compact and versatile. The kit includes a power adaptor, though its not switched.