Providing customers with flexible and uncertain due dates from the perspective of the supply chain system can improve the competitiveness of the supply chain and customer satisfaction. At the same time, it also brings challenges to the efficient operation of the supply chain system. To this end, this paper considers the characteristics of collaborative enterprises in the supply chain collaborative manufacturing model, such as different production startup costs, the need for advance preparation, and long procurement lead times. A two-stage supply chain collaborative robust scheduling optimization model is constructed based on the box uncertainty set of due dates. This model is designed to minimize the costs of the supply chain system by reducing decision conservatism. The column-and-constraint generation algorithm (C&CG) is used to solve this model exactly, and the subproblem is transformed based on the convexity of the objective function under uncertainty to construct a simplified dual problem. Numerical simulation examples analyze the impact of uncertain due date changes on the two-stage robust scheduling strategy and compare the robust optimal cost of single-stage, two-stage, and hindsight strategies under different real due date scenarios. Meanwhile, the real cost due to the deviation between the estimated and the actual due date intervals is compared and analyzed. The results show that the cost of the two-stage robust model is significantly lower than that of the single-stage robust model, and the gap is also not large compared to the hindsight optimal cost. In addition, the cost of the two-stage robust model shows relatively small variance, and the two-stage robust solution is much less sensitive to the estimation error of the due date intervals than the single-stage robust solution. Therefore, the two-stage robust optimization model can significantly improve the conservatism and inflexibility of the scheduling strategies and effectively respond to the needs of flexible, uncertain due dates.