I did not find a design to replace the broken rear hinge of the Logitech C270 webcam. Mine broke so I made a new hinge that's meant to superglued on. I used an STP file of another maker (Voron Trident C270 Mount by Ibot_3D | Download free STL model | Printables.com) to design a new rear hinge piece.

Sand the rear panel of the case flat and rough and possibly sand the mating face of the printed part a bit aswell for better adhesion. Apply superglue by following its instructions and voila, you have yourself a new and stronger hinge with a better offset for variable mounting options.