Error connecting to all of my SQL servers

The question seems to have been answered, but I wanted to chime in. For some providers, such as SQL Server, there is a parameter in connection string which lets you connect to server encrypted even if certificate is unknown: "TrustServerCertificate=True", so if you include that in a connection string, you will connect and work encrypted, and will not have to run connection non-encrypted.

Try this...

Its gotta be a client issue if you lost connection to all your remote servers and your coworkers are fine. You probably got "clicky" and changed some settings inadvertantly.

Open your client network utility (mine is here: C:\WINDOWS\system32\cliconfg.exe). Under the General Tab, check out the disabled protocols. They should all have "force protocol encryption" unchecked. If this is checked for any of those values, your local SSMS is probably trying to force an encrypted connection and failing.

Report back if this doesn't work, and I'll poke around a bit more.