I decided to test additional versions of Wine with the Windows version of Steam via Lutris. Here are the results:
I can confirm seeing this. The server itself is working OK, I could connect to it from a Windows installation.
The reason for this failure to connect through Proton I think it has more to do with recent changes in the game than Proton itself. At this moment I can connect without issues to Rust stable servers (eac disabled) with various version of Proton.
I can only assume that this is happening because they switched to IL2CPP instead of Mono in their staging build.
Whatever the reason is, the fact that this issue cannot be reproduced when running the Windows version of Steam with any non-Proton wine version gives me hope that the fix should be simple, once the root cause has been identified.
proton 3.7x1 2019-08
Compatibility Report
System Information
I confirm:
steam-700580.log
Symptoms
The client is unable to authenticate. As a result, it is not possible to join any Rust Staging Branch servers. However, this same issue does not occur when launching Windows Steam from Lutris using vanilla Wine 4.14. Using Wine 4.14 to launch the Windows version of Steam, I can successfully join Rust Staging servers. It's also worth noting that this issue is only present for Rust Staging servers and not normal Rust servers. However, this may change in the near future (as early as 09/05) as many features from Staging make their way into the main branch.
When testing with Proton 3.7, I encountered a different (but possibly related) issue immediately upon launching Rust Staging: "Steam Load Error - Is Steam Open? Please exit and try again with Steam open."
Reproduction
At this point, you should be looking at the error "Steam Auth Failed".