Repairing a broken screw hole on a plastic junction ceiling box?
That box is a type of plastic called Bakelite. It's tough, but brittle. If you're careful, you can break the box apart without damaging the ceiling. If you have a hole saw, you can probably drill out around the rivet heads you see in the back of the box and it will come right out.

Then you can replace it with a new "old work" box.