No, GhostBed mattresses don’t require a box spring, but you may use one.
What matters most is to have a sturdy foundation, such as an adjustable base.

This includes most platform, slat, and box spring foundations. For slat foundations, check to make sure slats are no more than 3-4″ apart.
Alternatively, add a bunkie board to your bed frame for extra support or use a low-profile box spring.