Another answer here also mentions since 2012 you can remove Remove cached login via How to remove cached server names from the Connect to Server dialog?. Just confirmed this delete in MRU list works fine in 2016 and 2017.
SQL Server Management Studio 2017 delete the file
C:\Users\%username%\AppData\Roaming\Microsoft\SQL Server Management Studio\14.0\SqlStudio.bin
SQL Server Management Studio 2016 delete the file
C:\Users\%username%\AppData\Roaming\Microsoft\SQL Server Management Studio\13.0\SqlStudio.bin
SQL Server Management Studio 2014 delete the file
C:\Users\%username%\AppData\Roaming\Microsoft\SQL Server Management Studio\12.0\SqlStudio.bin
SQL Server Management Studio 2012 delete the file
C:\Users\%username%\AppData\Roaming\Microsoft\SQL Server Management Studio\11.0\SqlStudio.bin
SQL Server Management Studio 2008 delete the file C:\Users\%username%\AppData\Roaming\Microsoft\Microsoft SQL Server\100\Tools\Shell\SqlStudio.bin
SQL Server Management Studio 2005 delete the file – same as above answer but the Vista path.
C:\Users\%username%\AppData\Roaming\Microsoft\Microsoft SQL Server\90\Tools\Shell\mru.dat
These are profile paths for Vista / 7 / 8.
EDIT:
Note, AppData
is a hidden folder. You need to show hidden folders in explorer.
EDIT:
You can simply press delete from the Server / User name drop down (confirmed to be working for SSMS v18.0). Original source from https://blog.sqlauthority.com/2013/04/17/sql-server-remove-cached-login-from-ssms-connect-dialog-sql-in-sixty-seconds-049/ which mentioned that this feature is available since 2012!