Application has been made of first-principles total-energy band-structure theory to find the equilibrium states under constant pressure of the super-hard cubic diamond (cd) and hexagonal diamond (hd) structures of carbon (C), boron nitride (BN) and medium-hard silicon (Si). The absolute stability of the equilibrium state is found by determinations of the breakdown under pressure of several deformations of lattice parameters around the equilibrium state. The calculations show that the hd structures are much stronger than the cd structures. Thus the γ angle of the hd structure of both C and BN is stable for pressures greater than 20 Mbar while the γ angle of the cd structures breaks down at 13 and 11 Mbar respectively. Also the bulk moduli B of the hd structure of C and BN are substantially greater than the B values of the cd structure above 2 Mbar; the B values of hd structures of C and BN are 20% greater than cd structures at p = 20 Mbar. However the cd structures have greater stability relative to the hd structures as shown by a lower Gibbs free energy at pressures up to 20 Mbar. Comparison is made with the pressure dependences of the medium-hard crystals of Si in the same structures, which show notably different behavior.