If you want remove connections from login list in Microsoft SQL Server 2014. Go to folder:
C:\Users\'WindowsUser'\AppData\Roaming\Microsoft\SQL Server Management Studio\12.0\SqlStudio.bin
WindowsUser - name of user in system Windows. Delete file SqlStudio.bin. Then your list of connections will be empty. How to delete a specific connection line - remove specific text from file SqlStudio.bin.