Data compression has the potential to significantly improve the computation offloading performance in hierarchical fog-cloud systems. However, it remains unknown how to optimally determine the compression ratio jointly with the computation offloading decisions and the resource allocation. This joint optimization problem is studied in the current paper where we aim to minimize the maximum weighted energy and service delay cost (WEDC) of all users. First, we consider a scenario where data compression is performed only at the mobile users. We prove that the optimal offloading decisions have a threshold structure. Moreover, a novel three-step approach employing convexification techniques is developed to optimize the compression ratios and the resource allocation. Then, we address the more general design where data compression is performed at both the mobile users and the fog server. We propose three efficient algorithms to overcome the strong coupling between the offloading decisions and resource allocation. We show that the proposed optimal algorithm for data compression at only the mobile users can reduce the WEDC by a few hundred percent compared to computation offloading strategies that do not leverage data compression or use sub-optimal optimization approaches. Besides, the proposed algorithms for additional data compression at the fog server can further reduce the WEDC.