The structural, mechanical, electronic, and magnetic properties of the new series of Co-based Equiatomic Quaternary Heusler Alloys (EQHAs) XX'YZ (X = Co; X' = Zr, Ru; Y = Ti, V; Z = Sn, Al, Ga, Si, Ge) have been calculated using first-principles calculations. The Full Potential-Linearized Augmented Plane Wave (FP-LAPW) approach was used in association with Density Functional Theory (DFT). We treated the exchange-correlation (XC) energy functional using a Generalized Gradient Approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) scheme. We included the TB-mBJ potential (GGA-mBJ) and the Hubbard correction (GGA + U) to improve the precision of the band gap values. Spin-orbit coupling (GGA-SOC) calculations are required due to the existence of heavier elements such as Zr and Ru. We have included the GGA-SOC calculations to estimate the total and partial magnetic moment and their impact on the electronic structure of Co-based EQHAs. The ground state parameters of proposed Co-based EQHAs were determined, including the lattice parameter, total energy, bulk modulus, and its derivative. The electronic structure of these materials revealed the presence of half-metallic ferromagnetism and spin-gapless semiconductor behavior. In addition, the origin of the band gap in spin channels was also investigated. In addition, we employed the mean-field approximation (MFA) method based on the Heisenberg model to determine the Curie temperature (TC). This theoretical analysis adds a new level of sophistication to the application of Co-based EQHAs in spintronic applications.