Found my answer which I already knew but I was just being stubborn!

http://msdn.microsoft.com/en-us/library/microsoft.sqlserver.dts.runtime.connectionmanager.connectionstring.aspx

It says:

The format and contents of the connection string depend on the data provider, the data source, and the type of connection manager that is being created. To view a sample connection string, in SQL Server Data Tools, create a package, and then create a connection manager of the appropriate type within that package. Next, in the Properties window, examine the ConnectionString property of the connection manager that you just created.

But my expression now looks like this:

"Data Source=" + @[User::Server_Name] + ";User ID=" + @[User::User_Name] + ";Password=" + @[User::Password] + ";Provider=SQLNCLI10.1;Persist Security Info=True;Auto Translate=False;Application Name=SSIS-Gather_Server_and_Database_Information-{30870F9F-B4A8-4EFF-B6C9-0C2DFB43480E}" + @[User::Server_Name] + @[User::User_Name]+ ";"

What's all this Application Name=SSIS-Gather_Server_and_Database_Information-{30870F9F-B4A8-4EFF-B6C9-0C2DFB43480E} nonsense?!