It is not easy physically to change a hard drive in the Mac Min (2012), or perhaps better: it is not for the faint of heart. There is, however, a way of change your boot drive to an external SSD by just changing the Startup Disk in System Preferences. I have done this for a number of years. My latest configuration is a Vertex 3 Series VTX3-25SAT3-120G connected to a Seagate Thunderbolt adapter (Model: STAE128) and a Thunderbolt cable to the Mac Mini (or to an Apple Thunderbolt Display).
The computer boots up very quickly and you never see Apple's equivalent of the hour glass after setting up the drive as your system drive. An internal SSD might be slightly faster, but I doubt this would be noticeable.
The ability to specify any disk as a start up disk is one of the strengths of OS X. I just hope that it will not be axed in the future so that the borders of the Apple Ecosystem are closed even more to third-party "infringements."