MnBi₂Se₄, MnSb₂Se₄, and MnSb₂Te4 are layered magnetic quantum materials of interest for spintronic and topological electronic applications, where control of magnetic order at the atomic scale is essential. These materials are particularly sensitive to strain. We investigate their strain-dependent magnetism in the monolayer limit using first-principles calculations across the full two-dimensional strain space spanned by independent variations of the two in-plane strain components. Our results show that the Mn-projected local moments depend primarily on the volumetric strain, whereas the magnetic ground state, exchange interactions, magnetocrystalline anisotropy, and ordering temperature show pronounced directional and composition-dependent responses. We also observe strain-induced competition between the phases, with MnBi₂Se₄, and MnSb₂Se₄ monolayers exhibiting a more diverse range of competing magnetic states, while MnSb₂Te₄ monolayer remains ferromagnetic over a larger portion of strain space. MnSb₂Te₄ also exhibits a substantially larger out-of-plane magnetocrystalline anisotropy, while showing a weaker dependence on strain than the Se-based materials. The calculated magnetic ordering temperature maps further highlight the role of two-dimensional strain to tune thermal magnetic stability in these materials. These results establish magnetoelastic trends across this family of Mn-based chalcogenide monolayers.