Funny thing is, on a Mac I think you have a name for the local computer and a name for the hard drive instead of a drive letter designation. Would it work on a Mac, I wonder?
If you had to have a UNC to the local machine you could always share your C: drive as 'mymachine' and grant security priviledges only to yourself. Then you would have a network redirector to provide the UNC, and \\mymachine\c would work.