In recent years, integrated quasi-zero-stiffness (QZS) vibration isolation and energy harvesting structures have attracted extensive attention and found wide applications in the field of low-frequency vibration isolation and energy harvesting for engineering structures. However, traditional methods for achieving QZS are typically limited to using linear springs to provide positive stiffness and bistable structures to provide negative stiffness. This paper proposes a novel dual-nonlinear combination method for realizing QZS, conducting a systematic theoretical study on the related principles. By designing a special negative stiffness structure combined with a hardening positive stiffness structure, a QZS system is formed. A verification-oriented integrated QZS vibration isolation and energy harvesting system is constructed to validate the effectiveness of this dual-nonlinear stiffness combination method fully. This verification system utilizes nonlinear negative stiffness inspired by negative Poisson's ratio structures, combined with nonlinear positive stiffness provided by repulsive magnets, to form an asymmetric QZS system. Static analysis reveals the compositional mechanism of its QZS, while dynamic analysis demonstrates that the proposed QZS system exhibits a low resonance frequency and a broad vibration isolation frequency band, enabling low-frequency vibration isolation and vibration energy harvesting. The proposed dual-nonlinear QZS structure can be used to develop integrated low-frequency vibration isolation and energy harvesting systems for vibration isolation and monitoring in engineering structures.