Auxetic mechanical metamaterials have extensive applications in energy harvesting and absorption; however, their mechanical properties still require improvement. This study proposes a novel two-dimensional auxetic mechanical metamaterial plate (NAMP) derived from conventional metamaterials. The NAMP offers excellent design flexibility and ease of customization, enabling rapid development of structures with superior auxetic behavior and stiffness nonlinearity for arbitrary dimensions. By adjusting its geometric parameters, a variety of mechanical properties can be achieved. Furthermore, a three-dimensional auxetic mechanical metamaterial tube (NAMT) was developed based on the NAMP, achieving the transition from two-dimensional to three-dimensional metamaterials. The NAMT not only exhibits favorable axial stiffness nonlinearity but also demonstrates significant anisotropy in Poisson’s ratio: under axial (Z-axis) tension, it expands along the X-axis while contracting along the Y-axis. This unique behavior broadens its potential applications. Overall, the mechanical properties of the NAMP and NAMT suggest promising applications in energy harvesting, absorption, and sensing technologies.