Auxetics occupy a special place among structural products working under cyclic loads, such as orthopedic implants, elements of building structures, etc. In this work, a three-dimensional auxetic structure (with Poisson’s ratio of −0.450) was proposed and the mechanical characteristics were analyzed using static and low-cycle compression tests (up to 2000 cycles) by both testing physical samples obtained by 3D-printing using SLM method and by running a computer simulation (FEA). The studied properties of auxetics were compared with the properties of a non-auxetic hexagonal honeycomb structure. It has been demonstrated that the auxetic structure is deformed uniformly during cyclic tests with a maximum load of 12 kN at a failure; accordingly, a non-auxetic structure loses stability at a load of 8 kN. The results can be used in the development of structural elements of biomedical devices operating under increased cyclic loads (spinal implants, bone implants for supporting limbs).