The mechanical response of single-walled zigzag silicon-germanium nanotubes (SiGeNTs) under tensile and compressive loadings is modeled via atomistic simulation method. The inter-atomic forces are described using the Tersoff's empirical bond-order potential. Initially, the comparative simulations of the bond lengths are presented and quite accurate behavior of the model is demonstrated. Afterwards, the results of the total strain energy are used to establish an expression for evaluating Young's modulus of the nanotubes. Since different choices of wall thickness significantly affect the calculation of Young's modulus, the effective modulus of elasticity is introduced. This procedure provides an accurate means for predicting the elastic modulus of the nanotubes. Numerical simulations were also performed to investigate the buckling behavior of SiGeNTs. Dependence of the critical buckling load on diameter and length of the nanotubes is shown. Finally, the effect of temperature on the axial compressive load of SiGeNTs is also discussed.