I tried a few things, but the only way I could silence the camera "click" was to turn off all sounds. I use a free app, MySettings to be able to do this quickly (MySettings>Ringer Mode>Silent).
I would have thought that there would be a way to do this in the program settings for the camera app, but I could find no way to do so (even tried Settings>Applications>Manage applications>All>Camera, to no avail).