This study investigates the influence of the deformability and fracture energy of the constituent material on the compressive response of auxetic cellular composites, using the finite element method (FEM) in ABAQUS/Explicit (version 2019). Four constitutive models were implemented: elastic-brittle, ideal elastic-plastic, strain-hardening, and strain-softening. The unit cell model was validated numerically against a larger 4 × 4 cellular structure and experimentally using strain-hardening cementitious composites with various deformability. Results show that auxetic behavior is unattainable with elastic-brittle constituent materials. For ideal elastic-plastic and strain-hardening materials, increasing the deformability and/or fracture energy leads to a larger critical strain, defined as the strain at which Poisson's ratio recovers from negative to zero under compression. Conversely, strain-softening materials exhibit the opposite trend. For structures comprising three ductile constituents, both load-bearing capacity and energy absorption performance improve with enhanced material properties, most notably for the strain-hardening material. However, a key finding is that increasing the deformability or fracture energy of the constituent material causes a significant reduction in the ratio of energy absorption of the structure to that of its constituent material. This indicates that merely enhancing the deformability and fracture energy of the constituent material does not guarantee improved energy absorption of cellular composites, demonstrating that optimal design of cellular composites requires a synergistic balance between the material and structure, rather than solely maximizing material properties. These insights provide critical guidance for designing high-performance auxetic cellular composites.