I have a late 2012 Mac Mini, but I don't use it for games. However, the machine initially came with a slow, 1 TB 5400 rpm hard drive. I replaced it with a much faster Samsung 840 Pro 256 gig SSD, and my "need for speed" was (and still is) easily satisfied. So, from an "external" point of view, you might want to have the SSD inside the machine when you purchase it.
Here are the various, most recent models:
http://www.apple.com/shop/buy-mac/mac-mini
However, it looks like SSDs inside that machine are PCIe-based Flash Storage, so you cannot upgrade it yourself, like I was able to do with my late 2012 model.