Newspaper. They're not going to ship it back to you in that box anyway, or even ship you back the same panel. They don't want to keep it either.
I thought you were an engineer...
Newspaper will not absorb any of the energy of the forces that the package will encounter. Newspaper is a no-go and any damage claim will be denied by FedEx or UPS due to insufficient packaging... forget about the post office.
Packing peanuts are useless because once they absorb the energy the first time, then they cannot absorb a second shock. There's many shocks when it rides on the truck.
Foam is the best option. Air pouches are the next best option and after that is bubble wrap. Make sure all void space is filled so the monitor cannot move around in the box.
The easiest way to get the supplies is to go to a local staples or UPS Store. The UPS Store has a pack and ship promise. If they pack the box and the item is damaged, they'll refund the charges and the cost of the item.
Even though I work for the corporate company, I'm honestly not sure how the UPS Store would handle a RS label with the pack and ship promise. I assume they'll just charge you a fee for the materials and to pack but that will be up to the individual franchise.