This manuscript presents a pioneering three-dimensional lattice structure that can simultaneously exhibit negative Poisson's ratio (NPR), negative compressibility (NC), and negative thermal expansion (NTE). The coexistence of the three negative indexes, whether in natural materials or in artificial structures, is extremely rare. The lattice unit cell integrates an auxetic egg-rack structure with a non-auxetic cage-like structure. Analytical expressions for the elastic constants of the unit cell are derived by using Euler-Bernoulli beam theory and subsequently validated through finite element simulations. The analytical results show that the lattice exhibits not only NPR, but also NC in a direction or specific areas as well as NTE in a direction, certain areas, or even throughout the entire volume when appropriately tailored geometries and constituent materials are employed. Furthermore, parametric analysis revealed that these properties can be adjusted within a broader range, encompassing negative and positive values, enabling diverse combinations. Metamaterials that possess multiple and adjustable negative properties enable the development of multifunctional devices capable of adapting to mechanical loads, hydrostatic pressures, and temperature fluctuations.