When the Squeezebox plays either an internet radio station or music service like Pandora, the connection is directly between the Squeezebox and the station or service. The term used by SqueezeCenter is "Direct Streaming". The SqueezeCenter server (or SqueezeNetwork) only works as a means of initiating and controlling the connection. It normally doesn't proxy the stream. When you shut down the server the connection remains.
There's an option under Settings > Player > [playername] > Audio > MP3 Streaming Method that can change this behavior for MP3 based streams. You can change the method from Direct Streaming to Proxied Streaming. I've never used it, but I would guess that it shuts down the stream when the server is shut down. My understanding is that WMA streams can also be proxied if you disable the native WMA streaming under Settings > Advanced > File Types > Windows Media.