8/2 @ 4:30pm - After a client / server update last night, it appears EAC is working properly again and none of the false bans have persisted. 

12:05pm EST - EAC has posted a response on the Facepunch forums:

The SteamAuth service triggered a glitch in our system that caused some players to be disconnected and then be reported as banned when trying to join the server again, without actually being listed on our banlist. We've sorted that out during this morning and are now waiting for the game servers to receive an update that will revert the cached banlists.

No actual bans were placed.

Looks like this will all be sorted out once Rust releases an update. In the meantime, we'll keep EAC off on our servers.

12:01am EST - The 3rd party software which Rust uses to catch hackers, EasyAntiCheat, has had several huge ban waves in the past 24 hours. Unfortunately, it appears these waves may have taken innocent bystanders with them. Although false bans are hard to confirm, the number of reports mixed with odd server behavior leads me believe something screwy is going on here.

Looking through our logs, these bans appear to come shortly after timeouts with the Steam Authentication servers. We're also seeing some console spam to the tune of "User XXXXXXXXXX is already banned." 

I've pinged Garry and EAC, along with toggling off EAC on our servers for the time being. This will allow players who may have been wrongly banned to join again and hopefully stop any further false positives from occurring until more light is shed on this.

If you are a server owner and wondering how to toggle EAC off, the console command is: server.secure 0