The hexagonal quasi-zero-stiffness and star-shaped negative Poisson’s ratio metamaterial structures, with their advantages such as simple structure, convenient for post-assembly, having high static stiffness and low dynamic stiffness, provide new ideas for lightweight and efficient nonlinear vibration isolation structures in the aerospace field. Based on the principle of energy shielding, the compression buckling behavior of hexagonal quasi-zero-stiffness unit cell structure under different central circle diameters is studied. The finite element analysis results show that the central circle diameter has little influence on the critical buckling load of the unit cell structure, but the horizontal deformation for the left and right sides of the hexagon is larger, and the internal struts are prone to asymmetric deformation, which may result in serious instability. To improve its mechanical properties, a central star-shaped negative Poisson’s ratio structure is introduced to replace the central circular structure, and connecting rods in the horizontal direction are added. The analysis and test results show that the deformation of the improved unit cell structure is more reasonable. The transverse connecting rods make the unit cell structure more approximate to zero stiffness and get a higher buckling load, but the quasi-zero-stiffness range becomes narrower.