I did some further testing on this, and here are my findings:
1. I set the external app link to launch a different application. That one continues to work after restarting JRiver, or restarting the whole system.
2. I created a batch file to launch firefox.exe, and that has the same behavior as when I point directly to the location of the firefox.exe file. All works fine until I restart JRiver, or the computer. The batch file did bring up some problems with focus, and returning to jriver after closing firefox. However that is a moot point, since it's not really proven to be a solution yet.
3.The other button I created holds fine. It's a quick button to launch into live TV. It was created as an external app launch as well, pointing to c:\windows\system32\MC22.exe w/ command line arguments of /mcc 30002. That one works continually after a restart of JRiver or the system.
For the life of me, I can't figure out why it works and retains the setting with another app in the same directory as firefox.
This installation is on one machine, like I said, I'm new to this. There is definitely the possibility that I have configured something wrong in set up.