In recent years, disordered photonic materials with hyperuniform characteristics have attracted a great interest due to their unique optical and electronic properties. These materials can find applications in components that are used to develop next generation technologies such as high-performance transistors, optical sensors, etc. In this work, we explored the relationship between hyperuniformity and photonic band gap (PBG) properties of 2D structures composed of discs and squares with varying packing fractions (φ) and compositions. Our recent simulation studies with a mixture of squares and discs and a pure system with rounded squares revealed two novel phases, mosaic and polycrystalline, that are formed by varying the composition (or degree of roundedness) of squares. These phases are partially ordered phases that have microdomains of both hexagonal and square symmetries that coexist, and they seamlessly bridge regions where these geometries dominate. By carrying out the hyperuniformity analysis, we confirmed that the large-scale density fluctuations in the mosaic phase is suppressed relative to isotropic structures, indicating characteristics like perfect crystals and quasicrystals. We observed that the packing fraction had a significant effect on the photonic band gap in these structures and analyzed its correlation with the width of the band gap. We found the mosaic phases to exhibit larger transverse electric (TE) and transverse magnetic (TM) band gaps than isotropic structures at the same packing fraction. Overall, our study provides new insights into the properties and potential applications of nonlinear hyperuniform materials and highlights the importance of understanding the hyperuniformity and photonic band gap in disordered structures of squares and discs.