The mechanical properties of endofullerenes have been investigated by performing compression tests using finite temperature first principles molecular dynamics calculations. We considered various X@C60 systems, with X a single noble gas atom (He, Ne, Ar, Kr, or Xe), small molecules (H2O, CH4), or small helium clusters. In the absence of compression, it is observed that there is no or at best a negligible effect of X on the properties of C60. The compression simulations revealed several original findings. First, the influence of X on the stiffness of X@C60 can be quantified, although it is at most 12% for the studied cases. Next, both energy and contact force variations as a function of strain are demonstrated to depend on X. However, this is not the case for the yield strain and for the failure mechanism of the C60 shell. Finally, it is shown that the X@C60 compression could bring X to be in a high stress state. In the specific cases of H2O and CH4 molecules, a mechanism of stress assisted dissociation is observed.