Soft robots are well-suited for human-centric applications, but the compliance that gives soft robots this advantage must also be paired with adequate stiffness modulation such that soft robots can achieve more rigidity when needed. For this reason, variable stiffening mechanisms are often a necessary component of soft robot design. Many techniques have been explored to introduce variable stiffness structures into soft robots, such as pneumatically-controlled jamming and thermally-controlled phase change materials. Despite fast response time, jamming methods often require a bulkier pneumatic pressure line which limits portability; and while portable via electronic control, thermally-induced methods require compatibility with high temperatures and often suffer from slow response time. In this paper, we present a magnetically-controlled stiffening approach that combines jamming-based stiffening principles with magnetorheological fluid to create a hybrid mechanical and materials approach. In doing so, we combine the advantages of fast response time from pneumatically-based jamming with the portability of thermally-induced phase change methods. We explore the influence of magnetic field strength on the stiffening of our magnetorheological jamming beam samples in two ways: by exploiting the increase in yield stress of magnetorheological fluid, and by additionally using the clamping force between permanent magnets to further stiffen the samples via a clutch effect. We introduce an analytical model to predict the stiffness of our samples as a function of the magnetic field. Finally, we demonstrate electronic control of the stiffness using electropermanent magnets. In this way, we present an important step towards a new electronically-driven stiffening mechanism for soft robots that interact safely in close contact with humans, such as in wearable devices.