Even though users couldn't log into other servers because we had them locked down, they can still see them if they browse the network.
After doing some research I found a group policy Administrative template (ADM) file that I was able to use to remove the network from the left side of the file explorer but this didn't stop users from being able to type in network up in the address bar, which would then allow them to browse the network to see other local computer systems.
I did some further digging and found this site Which had information about how to edit the registry to remove the ability to see the network tab, but as an added bonus it also removes the ability to view any computers on the network if you change the registry key.
After testing it on a test system I knew it would work, but I didn't want to have to manually edit the registry or use a reg file on every single server that we host.
Since I had previously used the other ADM file I knew it was possible to make a custom file that could be imported into active directory to make the registry change, so I started researching on how to do this.
After reading up on how to create my own ADM file I created a file that gives the ability to edit the file explorer's ability to browse the local network and have verified that it does work.
If the user types in network on the address bar the only option that shows up is network (Domain Name). If they select this it shows entire network, and if they click this it shows a blank page with nothing, so they can't see other computers on the local network. Please note that this does not remove the ability to directly access a computer on the network by using a UNC path like
Again this wasn't an issue for us, due to how the servers are locked down.
For their environment the servers are server 2012, or server 2012 R2 depending on when they were setup, and this ADM file worked for both after I imported the template and enabled it. The ADM file may also work for windows 7 but I haven't tested it. My recommendation would be to review the site I provided the link for and check the registry keys for windows 7 or 8 and see if they are present and if the default entry for the registry key is the same. If so then this should work for you.
If you want to try my ADM file here is the