More important than packaging them properly is to make sure that you have insurance on them that covers damages.
When shipping big items, you want to talk to either the postal insurance or a third party insurance company and see what they require to give you complete damage protection during shipping. Usually the insurance company will give you a mimimum packaging requirements list with the insurance.
As for the questions itself, you wanna make sure that the legs/back rest do not break. That means bubble wrap around all parts, followed by trash newspaper and tape… That is how I got my stuff online and it came in perfect shape. The shipper had offered optional insurance and I took it of course. Better safe than sorry.
It isn’t cheap, but if you value ths chairs then The UPS Store should be able to meet your needs to pack and ship them.