I'm not sure why Norton complained with the 32-bit version. It just said that the program was trying to go through the firewall and didn't have a valid digital signature. I can follow up with Norton if you want, but I don't know if it is worth it.
Norton has a tool named File Insight that tells the publisher of a file, a general idea of how many Norton users are using that file, and whether Norton thinks the file is "safe" or not. Norton gave both the 32-bit and 64-bit versions of the file a "Good" rating, apparently indicating that they trust that the file is from JRiver and that JRiver is trustworthy. File Insight said that hundreds of Norton users are using the latest version of the 64-bit file but only a few Norton users are using the 32-bit version. Perhaps nobody else ran into the issue (which would be odd; you'd think the experience with Norton would be the same across users) or just didn't decide to report it.
Anyway, I'm running fine with the 64-bit version. Unless you want me to follow up with Norton, I'll consider the issue resolved. It seems that most JRiver/Norton users are using the 64-bit version and wouldn't see the issue anyway.
Thanks for your help, everyone!