Eventually, yes but I don't think that it has much to do with the sun. Jupiter's magnetosphere shunts all of the solar wind (plasma) from the sun and
Jupiter's atmosphere was also found to be quite turbulent. This indicates that Jupiter's winds are driven in large part by its internal heat rather than from solar input as on Earth. -nineplanets.org
Presumably, Jupiter will freeze some day regardless of sunlight (having assumed the sun is immutable). I would compare the amount of energy Jupiter is receiving from the sun to the heat you add to your coffee by stirring it with a spoon (friction). This is probably a gross understatement but it's one way to look at it.
The magnetosphere of Jupiter is the cavity created in the solar wind by the planet's magnetic field. Extending up to seven million kilometers in the Sun's direction and almost to the orbit of Saturn in the opposite direction, Jupiter's magnetosphere is the largest and most powerful of any planetary magnetosphere in the Solar System, and by volume the largest known continuous structure in the Solar System after the heliosphere.
Jupiter's internal magnetic field is generated by electrical currents in the planet's outer core, which is composed of liquid metallic hydrogen. Volcanic eruptions on Jupiter's moon Io eject large amounts of sulfur dioxide gas into space, forming a large torus around the planet. Jupiter's magnetic field forces the torus to rotate with the same angular velocity and direction as the planet. The torus in turn loads the magnetic field with plasma, in the process stretching it into a pancake-like structure called a magnetodisk. In effect, Jupiter's magnetosphere is shaped by Io's plasma and its own rotation, rather than by the solar wind like Earth's magnetosphere. Strong currents in the magnetosphere generate permanent aurorae around the planet's poles and intense variable radio emissions, which means that Jupiter can be thought of as a very weak radio pulsar. Jupiter's aurorae have been observed in almost all parts of the electromagnetic spectrum, including infrared, visible, ultraviolet and soft X-rays.
The action of the magnetosphere traps and accelerates particles, producing intense belts of radiation similar to Earth's Van Allen belts, but thousands of times stronger. The interaction of energetic particles with the surfaces of Jupiter's largest moons markedly affects their chemical and physical properties. Those same particles also affect and are affected by the motions of the particles within Jupiter's tenuous planetary ring system. Radiation belts present a significant hazard for spacecraft and potentially to human space travelers. -Magnetosphere of Jupiter, Wikipedia
There are a lot of variables; Jupiter is its own mini solar system. Humans have neither the means nor the time to study the effects of it otherwise (I.e, take away its moons, negate the sun or put Jupiter alone in a box for billions of years) and I wouldn't trust any simulation for a definitive answer. The WISE object that Phil mentions could be compared to a rogue planet, and it is surprisingly similar in size (however its mass is 3~10 times larger). Jupiter Mass, Wiki:
If the sun had the capability of keeping Jovian planets from freezing I'd expect Mercury to be completely molten. Given the proximity of Mercury and Jupiter to the sun I believe tidal forces and convection to be the much larger factors at play here than sunlight. The point is moot. You can't live on gas giants anyway and even if it was frozen solid, 2.5x normal gravity would not be pleasant (and a bit nippy). The chances of a rogue planet entering our solar system and achieving Goldilocks status are, wait for it... astronomically [echo] slim.