How do I create a shortcut that opens Cygwin bash shell on a directory of my choosing (UNC especially)?

I haven't toyed with UNC paths, but in general you have to use forward slashes instead of backslashes I believe. For regular drives, you can use /cygdrive/c for example to access the C drive.

In a batch file, to open the shell to the root of C: you could use:

c:\cygwin\bin\bash.exe --login -i -c 'cd "/cygdrive/c";bash'

You may also be interested in chere a context menu entry for Open bash here:

I added a subfolder in registry at HKEY_CLASSES_ROOT/Directory/shell, added key Cygwin with key command where the command's value is

cmd.exe /k "E:\PATHTOCYGWIN\bin\bash --login -i -c "cd ""%L"";bash""