Install older OS X on latest Mac Mini
You can not install older OS X versions on newer models, the old OS X version are missing the required drivers for hardware not even available when the OS X version was released.
I see several options:
You may (legally) install 10.6.8 Server in a VM (Parallels or VMware) though.
The latest MacMini supporting 10.6.8 is the MacMini4,1: Apple Mac mini "Core 2 Duo" 2.4, Apple Mac mini "Core 2 Duo" 2.66 & Apple Mac mini "Core 2 Duo" 2.66 (Server)
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With some workarounds and hazzle you will get 10.6.8 working on MacMini5,1 and MacMini5,2. In the comments there (March 12, 2013 Gregg Tomlinson) even a working MacMini6,1 is mentioned:
I believe it does – I am using this on a 2012 MacMini that came with Snow Leopard [sic!], and reverse-engineered it to use 10.6 with this methodology.
While you can't run an older Mac OS X version on hardware that predates it (driver issues) if you must run Snow Leopard on newer OS X hardware consider installing it under the free VirtualBox virtual machine. It runs Snow Leopard and Snow Leopard Server just fine, albeit slightly slower. Make sure you have adequate RAM.