The aspect ratio has a significant effect on buckling instability. In this study, we used molecular dynamics (MD) simulation to investigate the buckling instability mechanism of NiCoCr medium-entropy alloy (MEA) nanotubes under the compression process. Based on the compression results, we find that the buckling point and the critical buckling stress decrease with increasing aspect ratio. The critical buckling stress of the MD simulations resembles the Euler and Rayleigh-Ritz theories. The critical aspect ratio of 10 showed instability at NiCoCr nanotubes. In addition, we found that the compressive strength of single-crystalline specimens is much higher than that of polycrystalline specimens. Partial dislocations (PDs) dominated by stacking faults (SFs) and twinning structures are mainly distributed on the grain boundaries of small grain specimens and decrease with increasing grain size. We also found that polycrystalline structures tend to induce buckling instability. The wall thickness specimen of the 7 nm has confirmed similar compressive strength as the nanowire. The compressive strength decreases as the wall thickness decreases. More stress concentration and PDs distributed in the specimen when the wall thickness is reduced to 1 nm, and it is easier to induce buckling instability at NiCoCr MEA nanotubes.