The problem was that I wanted to move the database from one server to another, but once it got to the new server I couldn't connect from anywhere except the server itself.
The error message was:
The company could not be opened. Please ensure the SQL Server exists, the service is running and access has been granted.Like I said, I tried a ton of stuff, but when I finally noticed that it was working, I modified the multi-user SBC file to read:
SERVERNAME\MSSQLSERVER,5356
instead of just SERVERNAME,5356
That's the first and only time I've ever seen a named instance required when you only have one instance running, but it works now.