We study the assembly of single-patch colloidal Janus particles under steady shear flow via Brownian dynamics simulations. In the absence of flow, by varying the Janus patch size and the range and strength of the anisotropic interaction potential, Janus colloids form different aggregates such as micelles, wormlike clusters, vesicles and lamellae. Under shear flow we observe rearrangement, deformation, and break-up of aggregates. At small and intermediate Péclet (Pe) numbers - the ratio between shear and Brownian forces - the competition between rearrangement, deformation, and break-up favors the growth of micelles and vesicles increasing mean cluster size, which is consistent with a previous numerical study of Janus particles under shear. This initial shear-induced growth causes micelles and vesicles to reach a maximum cluster size at Pe ≈ 1 and Pe ≈ 10, respectively. After this growth micelles dissociate continuously to reach a dilute colloidal "gas phase" at Pe ≈ 10 while vesicles dissociate into micelles with high aspect ratio at Pe ≈ 10 and finally break-up into a gas phase at Pe ≈ 30. Wormlike clusters initially break-up into micelles with high aspect ratio at Pe ≈ 0.1, and proceed to finally reach a gas phase at Pe ≈ 10. Lamellae initially break into smaller lamellae that align with the flow in the velocity-velocity-gradient plane and finally break-up into a gas phase at Pe ≈ 100. The different cluster sizes and morphologies observed as functions of interaction range, Janus patch size, interaction strength, and shear rate, open new actuation routes for reconfigurable materials and applications.