It's apparently a bug in Opera 41. I uninstalled it to go back to 40.0.2301.0 and the VPN is working fine now.
To stop auto-updates, I changed the name of the executable file "opera_autoupdate.exe" located in C:\Program Files (x86)\Opera developer\40.0.2301.0 to "no_opera_autoupdate.exe."
Opera needs to fix the VPN bug, and add a setting to disable auto-updates to help prevent this kind of problem.