We performed the modified-embedded-atom-method (MEAM) based molecular dynamics (MD) simulations to investigate the plastic deformation and phase transformation behaviors in the CoCrFeMnNi HEA under high compression at room temperature. Our MD simulations revealed that the stress-induced phase transformations in the CoCrFeMnNi HEA are strongly crystal orientation-dependent. The [0 0 1] uniaxial compression can induce the significant face-centered-cubic (fcc) -to-hexagonal-close-packed (hcp) phase transformation via successive emissions of partial dislocations from the extended stacking faults, twin boundaries and hcp-lamellas created during the early stage of deformation. As for the [1 1 0] and [1 1 1] uniaxial compressions, however, the transformed hcp atoms can simply form the intrinsic/extrinsic stacking faults. Although the [0 0 1] uniaxial compression produced a much lower dislocation density than the other two systems, it induced much more constituents transformed into the hcp atoms at the end of phase transformation. Our results clearly indicated that the deformation twin boundaries and extended hcp-lamellas play a critical role in facilitating the stress-induced fcc-to-hcp phase transformation in the CoCrFeMnNi HEA. Furthermore, it was found that the phase transformation in the CoCrFeMnNi HEA can be effectively facilitated by a large deviatoric compressive stress while it may tend to be significantly retarded by a hydrostatic compression. Our results also showed that the plastic deformation behaviors in Ni under high compression are very similar to those occurred in the CoCrFeMnNi HEA though nearly all the hcp atoms can simply constitute the intrinsic/extrinsic stacking faults without the formation of any bulk hcp phase in the fcc lattice. The main discrepancy in the phase transformation behaviors between the Ni and CoCrFeMnNi HEA can be largely attributed to the much lower stacking fault energy of the CoCrFeMnNi HEA than other fcc metals.