From information online, those printers use regular Ink Jet ink to print on nails. A project like this would be pretty difficult to do IMO. I'm not sure exactly how the Ink head comes in contact with the nail, but the software side alone would be difficult to implement. You'd have to be a pretty good programmer unless there's some software I don't know of that could be modified. The Nailjet software has a camera that identifies the location and shape of the nails and then takes the artwork and generates an image to fit on the exact nail. So to come up with a system that could do that while generating the XYZ coordinates for the print head, figure out all the timing, come up with a way to precisely and accurately make a print head gently come in contact with a nail....each nail being curved and unique...I am not trying to deter you but it'd be pretty difficult and take a lot of knowledge, trial and error. It would be neat to see a system like this implemented because I can see the technology could be put to many other uses.
@Atomist your right about this tech being put to other uses. I think it would be cool to build a resin based ink jet 3D printer! A good start would be this cool Open Source ink jet head to hack InkShield