Zr₂SB, Hf₂SB, Zr₂SeB, Hf₂SeB, and Hf₂TeB are all recently discovered S-group elements containing MAX-phase borides, which attract much attention since the MAX phase borides are significantly unlike the typical MAX phases. Here, the phase stability, mechanical properties and thermal properties of MAX phase borides (M = Zr, Hf, A = S, Se, Te) were studied by using first principles and "linear optimization method", bond stiffness model and quasi-simple harmonic approximation. The results of the theoretical analysis were consistent with the currently available experimental results. Only M₂AB was found to be stable after thermodynamic and intrinsic stability analysis. The shorter M-A bond and M-B bond lengths cause bond stiffness of Hf lineage higher than that of Zr, which also leads to the higher hardness of Hf lineage compound than that of Zr. the A site element goes from S to Se and to Te, the bond lengths of M-B and M-A are gradually increased, which lead to decrease in the elastic modulus. Moreover, the bulk modulus of these compounds is determined by their average chemical bond stiffness. Importantly, the high k_min/k_max (stiffness ratio of the weakest and the strongest bonds) shows that these MAX phases are inherently brittle, different from conventional MAX phase. Including the contribution of lattice vibration (phonon) and electron excitation, the isobaric heat capacity and heat expansion coefficient of M₂AB increase rapidly with increasing the temperature below 300 K and then the rise rate gradually decreases, similar to other MAX phases. Lower bond stiffness results in an overall higher TEC of MAX phase borides in the Zr lineage than in the Hf lineage. The TEC values of these compounds in the 300-1300 K interval are consistent with most of the MAX and MAB phases.