BES Method Correct. When you run "BlackBerry Browser" through BES, you are using your work's Internet connection. That's because BES is usually on the Intranet (inside the LAN) and you can access internal websites from your BlackBerry that way. And yes your boss can see what websites you are accessing if you surf through BES -- if they have enabled the logging and someone is watching the log.

BWC Method If you need privacy, use "Internet Browser" instead of "BlackBerry Browser", it will surf through your BWC instead.