Okay, so the screws were a bit of a red herring. They'll come in handy for replacing a cracked lens cover or o-ring, but they didn't get us inside.
Prying at the front and back covers didn't get us anywhere either. So, we gave our new camera a hot iOpener hug to soften up the adhesive on its rubber cover—and then started cutting and peeling.
At the top of the camera, the shutter button is integrated into the rubber cover, with a hole in the plastic case underneath exposing the microswitch.