We have investigated the shear-induced ordering of multilamellar vesicles (onions) in a nonionic surfactant (C12E4) system using a small-angle light scattering (shear-SALS) and a small-angle X-ray scattering (shear-SAXS) technique. In a narrow shear rate-temperature space, the onions form a two-dimensional (2D) hexagonally close-packed structure that shows characteristic hexagonal scattering patterns in both SALS and SAXS. In the dynamic phase diagram, the ordered onion phase is surrounded by disordered onion phase, indicating reentrant behavior against the temperature. The disorder-order transition is accompanied by a jump in onion size by a factor of 5-6. In the disordered onion phase, by applying a shear flow, the planar lamellar membranes transform to an intermediate structure, multilamellae cylinders or a coherent stripe buckling, and then the intermediate structure develops to the isotropic onions. On the other hand, in the ordered onion phase, the intermediate structure breaks to onions stretched in the shear velocity direction, and then the stretched onions aligned gradually to form the 2D hexagonal close packing.