Can it be done - I think yes. Does it make any sense - I think no.
The concept of a non-chemical thermal rocket is not new. Other types of such rockets would be a nuclear thermal rocket or a resistojet. Both work in principle by heating a fluid and then expelling it, just like the concept you suggest. The fluid can be anything, but hydrogen is best in terms of ISP.
So how would a use case of your rocket look like? Obviously, you would not try to use it to escape earth orbit, because the energy dissipating at re-entry is just the motion energy you put in your spacecraft in the first place. Furthermore, most of the energy is not transferred to the spacecraft but stays in the atmosphere (if I remember correctly, it was about 0.2% going into the spacecraft).
But, as you were saying, maybe your want to use this energy to fly a maneuver in the Jovian system. When you enter the system, you have escape velocity, so you want to shed some of that energy. In this case by aerobrakeing in Jupiter (may not be smart for other reasons), yes, you may encounter some heat, and that heat, in theory, can be used as energy for later course changes. You could, for example pressurize a lot of hydrogen, then use it to cool a heat shield, and then release the hydrogen for thrust.
This would be a very bad design. You would have to store a large amount of pressurized hydrogen somewhere, because you don't want to fire your thruster right away, while still in Jupiter. You want to have propulsion hours, or days, or years later. This makes your tanks massively heavy. And that is just the least of your problems.
In order to get any sort of decent ISP, you would need to heat your hydrogen to thousands of kelvin. Therefore, it would not be an effective coolant. Actually, you would somehow have to actively cool your storage tank, because there is no material in the world that could resist the temperatures needed for a gas to be an effective rocket propellant.
And even that ignores the largest flaw in this concept, that energy isn't even the largest concern in spacecraft propulsion, it's momentum. There are many sources of energy available, even in outer space. There are radioisotope thermoelectric generators, solar panels, chemical reactions, even nuclear reactors. The tricky part is to use this energy to expel a fluid backward at the greatest speed (read: ISP) possible, so according to the momentum balance, your space ship gets the most forward momentum. In this regard, there is nothing better than ion thrusters, and your concept would do worse than the simplest thruster, the resistojet. This is a thruster that does nothing but heat something (anything really - hydrogen, water, poop, you name it) electrically to expel it through a nozzle. So just like your idea, the resistojet also also heats up the propellant, but it doesn't have to store it and it is not dependent on a specific part of the mission in order to function.
Therefore, even if it was possible to recover some energy from aerobrakeing, I don't think that it can be a viable part of any propulsion system.