This paper presents a novel hierarchical, safety-critical control framework that integrates distributed nonlinear model predictive controllers (DNMPCs) with control barrier functions (CBFs) to enable cooperative locomotion of multi-agent quadrupedal robots in complex environments. While NMPC-based methods are widely adopted for enforcing safety constraints and navigating multi-robot systems (MRSs) through intricate environments, ensuring the safety of MRSs requires a formal definition grounded in the concept of invariant sets. CBFs, typically implemented via quadratic programs (QPs) at the planning layer, provide formal safety guarantees. However, their zero-control horizon limits their effectiveness for extended trajectory planning in inherently unstable, underactuated, and nonlinear legged robot models. Furthermore, the integration of CBFs into real-time NMPC for sophisticated MRSs, such as quadrupedal robot teams, remains underexplored. This paper develops computationally efficient, distributed NMPC algorithms that incorporate CBF-based collision safety guarantees within a consensus protocol, enabling longer planning horizons for safe cooperative locomotion under disturbances and rough terrain conditions. The optimal trajectories generated by the DNMPCs are tracked using full-order, nonlinear whole-body controllers at the low level. The proposed approach is validated through extensive numerical simulations with up to four Unitree A1 robots and hardware experiments involving two A1 robots subjected to external pushes, rough terrain, and uncertain obstacle information. Comparative analysis demonstrates that the proposed CBF-based DNMPCs achieve a 27.89% higher success rate than conventional NMPCs without CBF constraints.