I have no idea where to get half a dozen egg cartons....
That is the easy part. You cut a dozen egg carton in half and you have 2 half dozen egg cartons.
The best packed eggs that I have received so far were tissue wrapped and placed in a half dozen egg carton. That carton was taped shut and placed in a small bubble wrap lined box. That box was taped up and placed inside a larger box which was line with bubble wrap and insulation. The insulation was there because the eggs were shipped during an extreme cold spell and had a 72 hour heat pack taped to the bottom of the small box.
The eggs arrived completely undamaged and could start hatching as soon as this Thursday.
Good luck.