Proximity-induced superconductivity in three-dimensional (3D) topological insulators forms a new quantum phase of matter and accommodates exotic quasiparticles such as Majorana bound states. One of the biggest drawbacks of the commonly studied 3D topological insulators is the presence of conducting bulk that obscures both surface states and low energy bound states. Introducing superconductivity in topological Kondo insulators such as SmB6, however, is promising due to their true insulating bulk at low temperatures. In this work, we develop an unconventional Josephson junction by coupling superconducting Nb leads to the surface states of a SmB6 crystal. We observe a robust critical current at low temperatures that responds to the application of an out-of-plane magnetic field with significant deviations from usual Fraunhofer patterns. The appearance of Shaphiro steps under microwave radiation gives further evidence of a Josephson effect. Moreover, we explore the effects of Kondo breakdown in our devices, such as ferromagnetism at the surface and anomalous temperature dependence of supercurrent. Particularly, the magnetic diffraction patterns show an anomalous hysteresis with the field sweep direction suggesting the coexistence of magnetism with superconductivity at the SmB6 surface. The experimental work will advance the current understanding of topologically nontrivial superconductors and emergent states associated with such unconventional superconducting phases.