I would think that you would want to make the front of the the box look like the soffit looked like before you hung the screen. With wall to wall Crown molding and hanging down even with the existing soffits.
Having said that the front face of this new box can be hung with screws along the top edge that get hid by the molding. Secure the molding with just a few finishing nails and caulk in place. If you need to access the screen remove the molding and unscrew the the front face.
You could put some small corner molding at the two side edges hiding any gap.