Does an all-foam mattress like this need a box spring?

No box spring is needed, and you should not use a sprung one. This 8-inch foam bed needs a flat, solid surface underneath: a platform bed, a bunkie board, closely-spaced slats no more than about 3 inches apart, or even the floor. A traditional box spring flexes and can let the foam sag prematurely. It is also rated adjustable-base compatible, bending smoothly with the frame since there are no coils to resist the articulation.