We consider the Inverse Optimal Stopping (IOS) problem where, based on stopped expert trajectories, one aims to recover the optimal stopping region through the continuation and stopping gain functions approximation. The uniqueness of the stopping region allows the use of IOS in real-world applications with safety concerns. Although current state-of-the-art inverse reinforcement learning methods recover both a Q-function and the corresponding optimal policy, they fail to account for specific challenges posed by optimal stopping problems. These include data sparsity near the stopping region, the non-Markovian nature of the continuation gain, a proper treatment of boundary conditions, the need for a stable offline approach for risk-sensitive applications, and a lack of a quality evaluation metric. These challenges are addressed with the proposed Dynamics-Aware Offline Inverse Q-Learning for Optimal Stopping (DO-IQS), which incorporates temporal information by approximating the cumulative continuation gain together with the world dynamics and the Q-function without querying to the environment. In addition, a confidence-based oversampling approach is proposed to treat the data sparsity problem. We demonstrate the performance of our models on real and artificial data including an optimal intervention for the critical events problem.