Does this memory foam mattress need a box spring, or what foundation works?

No box spring is needed. As a 12-inch all-foam mattress it wants a solid, flat, fully supportive surface: a platform bed, a slatted frame with gaps no wider than about 3 inches, or a bunkie board over an older box spring. A traditional sprung box spring flexes and can let the foam sag, so a firm flat base keeps the support core working as measured.