Does the game also not save any settings you set (keybindings, graphics options, etc..) or is it just your "progress" that isn't being saved?
While I'm not familiar with the Steam version of Squadrons (I have the Origin version), Steam usually holds save games in the \steamapps\userdata\<userId>\<gameId>\ folder. Have you ensured that you also have full read/write/ownership privileges on that folder? Could it be that it only saves settings in the Documents\Star Wars Sqadrons Steam\ folder?
Have you tried running the game on the full Administrator account (it's hidden by default and usually called "Administrator")?
How have you set up the documents folder on your network drive? Have you pointed Windows to the new location via Windows Explorer, have you manually created a symlink, hardlink or such? Perhaps try if a different way works better.
I know you don't want to hear it, but if nothing else helps, it could very well be that the game has problems accessing the network location. This could even be the case if other games don't have problems with it. Which file sharing protocol are you using to connect to the network drive? Perhaps changing it could work, especially if you use one that isn't native to Windows.You might want to try to set the Documents folder up on a local drive temporarily (just use explorer to set a new location for Documents and don't move it's contents) and see if it works then, just to rule out that it's the culprit (and if it works, try moving the Squadrons folder back to the Network drive and set Documents back to the Network drive and see if it works afterwards).
On a side note, I myself have the Documents folder set up on a storage drive, albeit it's a local HDD and not a network drive and (Origin version of) the game saves fine this way.