The quick short answer is, no. You do not need to loosen the strings of your guitar before selling or shipping, and here is why. If you have ever bought a guitar online, or even from a store, you would find that it is properly tuned.
Should you loosen strings before shipping a guitar?
The natural rigidity of the neck's wood is enough to handle the tension. So, you can safely loosen the strings prior to shipping, but you still need to pack the guitar adequately, and this includes padding INSIDE the guitar case as well as outside.
When should I loosen my guitar strings?
Should I loosen my guitar strings when not playing? It is not necessary to loosen your guitar strings when not playing. The guitar's neck can handle the strings tension in its tuned position when not playing, whether hanging on the stand or kept inside the case.