Hi Ryan,
If you are connecting to your geodatabase using 'Database Servers' then you are creating a direct connection. You cannot have an application server connection (use an SDE service) created through 'Database Servers'.
Do you have two ArcSDE licenses, one for Workgroup and another for Enterprise? A Workgroup license uses SQL Server Express for the database. With this type of license, you can create an administrative connection to your geodatabase through 'Database Servers'.
With an Enterprise license the ArcSDE Post-Installation is included. You can use the ArcSDE Post-Installation to create a service to connect the geodatabase, rather than use a direct connection.