There is no reason it should not work with a mini - it has DVI input, so I dont see any problems. The only thing is that you may not be able to use the 'portrait mode' advertised on the linked website, as i'm not sure if os x supports that feature - you'll have to get someone elses opinion on that one. Also, the quality of the screen will be slightly less of that on the apple 20" cinema display, as the gateway one has the same number of pixels as the apple one, but the gateways are slightly less dense due to the screen size, though it is only 1 inch, so the difference will most probably be un-noticeable.
The display should also work with any other mac, seeing as it also has a VGA input, which would have to be used if connected an iBook or an iMac.
Note: if the mini in question is an old model (ie the ones out now, with the 512 RAM and radeon 9200 with 32 VRAM), then expose and other graphic intensive stuff will be sluggish on a large display. If however, you manage to get a new bumped mini with 64 VRAM, then this should be improved