I finally went back and fixed the instructions at
http://jogamp.org/wiki/index.php/Building_JOGL_on_the_command_line to separately show how to install MinGW for 32-bit and 64-bit versions of Windows. The instructions at
http://jogamp.org/jogl/doc/HowToBuild.html have showed this correctly for a while now, I just kept forgetting to update the more detailed instructions to match.
I recently had to re-install 64-bit Windows on my box, so then I reinstalled MinGW using my own (wrong) instructions on the wiki and ran into the same problem on this thread just today. Pretty embarrassing :)