To build tissues, and ultimately functional bodies, cells in early embryos must arrange into specific patterns. In most animals, epithelial tissues exhibit a predominantly hexagonal space-packing geometry. However, in many species of the largest group of crustaceans, the Malacostraca, the embryonic epithelium takes on the striking form of a grid made up of predominantly square cells, sequential rows of which establish the adult segmented body plan. After square cells emerge, their organization appears to be maintained by specific cell division patterns. However, the mechanisms that initially generate square cells from hexagonal precursors are unknown. Here we address this problem by combining long-term multiview lightsheet microscopy, immunohistochemistry, laser ablation, and pharmacological perturbation. We show that in the emerging model crustacean Parhyale hawaiensis this highly unusual grid geometry is first initiated from two perpendicular axes that are established sequentially according to different cellular mechanisms. The first axis arises dorso-ventrally at a tensile lineage compartment boundary, while the second emerges at the ventral midline through lineage-independent cell intercalation driven by tensile actin-myosin cables. We show that these midline cables are necessary for organizing square-cell packing as well as for proper expression of the segmentation gene engrailed. Our findings show that both cell lineage-specific behaviors, as well as lineage-independent supracellular structures, are required to establish square grid epithelial organization necessary and a segmented body plan.