Most output-constrained methods necessitate reference command within a predefined safe region, without considering cases where the command itself may conflict with safety boundaries. To handle this problem, this article proposes a predefined-time safe cooperative control scheme for multiagent systems under output constraints, privacy preservation and unknown disturbances. At the communication layer, an encryption-decryption mechanism is developed to safeguard information exchange among agents, preventing internal states from being identified by eavesdroppers. At the control layer, to ensure strict adherence to output constraints regardless of whether the original command complies with safety limits, an improved boundary protection method is explored to generate a safety reference trajectory, which is subsequently used in the controller design. Adaptive laws are then formulated to counteract the effects of unknown nonlinearities and disturbances. Finally, by leveraging predefined-time stability theory, a predefined-time safe cooperative controller is designed to ensure error convergence within a user-defined settling time. Theoretical analysis rigorously confirms the closed-loop stability, and simulations verify the effectiveness of the proposed method.