Newer versions (starting from 5.1.7) of the Barracuda VPN client are also available on the Mac App Store. Upon installing via Mac App Store (or via direct download from Barracuda website), you'll notice that the connection profiles are gone and that only the standard default profile is present. This is because the new version looks for the configuration profiles in a different location so as to be compliant with requirements for apps on the App Store. Luckily, the old configuration files still exist and a simple relocate of the required file will bring back your profiles.
In older versions, everything about the Barracuda VPN clients, such as profiles, license files, etc. were stored directly in a hidden subfolder in your home directory: ~/.barracudavpn/. The file barracudavpn.conf in that folder contains the connection profiles. Other files will be present as well, such as certificate files or license files (with extension .lic).
To be compliant with App Store regulations, the configuration folder is now located in a group container, but inside the container, the folder structure is the same. By moving/copying the profile configuration file to the correct location within the Container will revive the profiles:
Before replacing the file, make sure the Barracuda VPN client isn't running. Then copy the file from the old location to the new, overwriting the existing file:
cp ~/.barracudavpn/barracudavpn.conf ~/Library/Group Containers/group.com.barracuda.Barracuda-VPN-Client/.barracudavpn/barracudavpn.conf
Optionally, if you want to be thorough and clean up the old folder, copy the remaining files and folders from ~/.barracudavpn/* to ~/Library/Group Containers/group.com.barracuda.Barracuda-VPN-Client/.barracudavpn/*. Note that if you do this, you will need to manually edit barracudavpn.conf in the new location and change any reference to certificate or license files to match the new location.
LICENSEFILE = /Users/user/.barracudavpn/license/Default.lic
LICENSEFILE = /Users/user/Library/Group Containers/group.com.barracuda.Barracuda-VPN-Client/.barracudavpn/license/Default.lic