Woo hoo! It's fixed!
First of all, thanks to Luis who helped me narrow it down to the point where I could figure it out. Luis, I took your advice and tried to run BR from the C: drive and then point to the network share after BR was connected. That worked! Yet, I could see the network share (with BR) via Windows Explorer and by popping down into the CMD line and doing an old-fashioned DIR. So I know that the drive was mapped correctly on the workstations.
Next I started cutting pieces out of my config files and determined that as I suspected, it died on the Drive statement, but depending upon what I screwed around with, sometimes I would get an error regarding the WBSERVER.DAT file. That gave me the final clue.
I asked the systems/network guy to check the permissions for the BR folder (I don't have authority to do that on their system). It looked good. Every user is part of the "Everyone" group and the Everyone group has full rights to the BR folder. But he refreshed the permissions and then it worked! I know they were having trouble with Windows patches, so it's possible that this could have been a cause of the permissions getting knocked out. But I don't know. The permissions were made for the "everyone" group. The people who couldn't get in were still listed in the group, but they weren't inheriting the permissions for some reason. After refreshing the permissions and pulling those people out of the Everyone group and putting them back in, It seems to be be working! We may never know what actually caused the permissions issue, but in the end, it had nothing to do with BR or my applications. No modifications of any kind ended up being required on my side of things whatsoever.
David had a good suggestion: don't depend on any built-in server "groups" from Microsoft to issue permissions. If MS creates them, they can mess with them. So we should create our own version of an "everyone" group and give permissions there instead. Good advice.
Thank you all so much!
And while I'm at it, there is a link available now from the Drury Plaza to book your hotel rooms for the May 19-21, 2016 Business Rules Conference. I'll go post it in the Conferences section in a bit.