When shipping, be sure to tape the puzzle's box closed. You may want to put the pieces in a ziplock bag inside the puzzle box if you are concerned that the puzzle box might open inside the shipping box. It is also a good idea to tape all openings of the shipping box. Not just the main flaps, but also the edges where the under flaps go so the tape is in an "H" pattern. This will prevent any puzzle pieces from slipping out if the inner puzzle box does pop open some how. This video shows an example on a large moving box but the concept works the same for puzzle boxes.
www.youtube.com/watch You may also choose to use brown shipping paper and simply wrap the puzzle like a gift, but make sure to fully tape the ends to protect the corners from ripping. This video shows the basic concept. You can adapt this for your puzzle box.
For the cheapest shipping within the United States, we suggest Pirate Ship. They offer the public business rate shipping. They make their money by keeping a few cents of the discount for each package that gets shipped using their site. They use the US Postal service and you simply put in the address and print out a label. Then you can ship directly from your house or just drop you package at the post office, usually without waiting in line. If we hadn't used them several times, we wouldn't be recommending them. We are not affiliated with them in any way, we are just happy customers. Visit their site here: ship.pirateship.com