The zip code needs to be character, for just the reason's you suggested.
Concerning your issue:
What version of JMP and what operating system are you running on?
Did you copy the 2 files to the directory specified? You may have to create some directories that are not currently available.
Both the -XY and the -Name data tables need to be copied to the correct folder
Since the zip code files are large, it may take a while for the map to show up.
Attached is a data table that I knows works with the zip code shape files, so if you try it, and it works, then the issue is with your data table. If you try it and it doesn't work, then your issue is with the install of the shape files from the Xan Gregg files I pointed you to.
Jim