Here is a trick that helps get the quilts 'into the box'! Put the quilt in a garbage sack. The white 13 gallon size works great.
Next, using your vacuum cleaner hose, put the tip of the hose in the sack and hold the plastic gathered up around it.
Suck all of the air out of the bag, you might have to even hold the hose tip against the quilt. When all of the air is removed and the quilt is 'flat', remove the hose and immediately twist the sack closed. You can then tape the twisted end down to hold it and immediately put the 'flattened' quilt into the shipping box. Tape the box closed at this point because over a few hours the quilt will grow back to its size---however by that time it is nicely packed inside the box.
Having the quilt in plastic for the short time it is in transit will not hurt and the plastic gives extra protection from moisture. I always include a note to the recipient of the quilt to not store the quilt in the plastic bag.