Do the magnetic fields of stars and/or other planets reverse?

Sure. It even happens with the Sun, as part of the solar cycle.

Earth's magnetic field is described by something called the dynamo theory, which states that the movement of fluids in the core generates the magnetic field. A modification of this is used to model the magnetic fields of stars. Given that the cause of geomagnetic reversals is related to the movement of currents deep on Earth's core, it makes sense that the same would be the case in other bodies.