How will the atmosphere deal with the extreme heat?
Using this article as a guide
During most of the red giant lifetime, the sun will be only 30 times
brighter than its current state. Toward the end of the red giant phase
the sun will grow more than 1,000 times brighter, and occasionally
release pulses of energy reaching 6,000 times current brightness.
Jupiter is about 5 times further from the Sun than the Earth, so it receives 1/25th the solar energy per area. If the Sun emits 30 times as much energy during most of its red-giant phase, and if we assume that Jupiter will migrate outwards some, that puts Jupiter (and its moons) probably somewhere in or close to the habitable zone.
Hydrogen and Helium are pretty inert, but Jupiter has enough Methane and Water vapor in its upper atmosphere to experience a greenhouse effect and gradually heat up, so it will likely grow hot given enough time, even at what we might consider a comfortable distance for a planet. The Red-giant stage of the Sun will last a couple hundred million years and Jupiter is very large and will take a long time to trap enough solar energy to really start to heat up, but I think that's the outcome. It would be a good place to put Earth during that time, but Jupiter would probably become a heat trap or run-away greenhouse planet at a certain point.
Even with heating up, Jupiter is massive enough that it probably won't lose much of its hydrogen.
A final point to make is that Jupiter will probably absorb a small percentage of the matter that the sun loses. Our Sun is estimated to lose about 54% of its mass by the time it becomes a white dwarf and much of (some of?) that matter loss will happen during the red giant stage. That's about 560 Jupiter masses.
Most of that ejected material will just form a planetary nebula or leave the solar system. A tiny percentage of it will be absorbed by Jupiter. I don't expect it will be much, but Jupiter will probably (could?) add some mass during the red giant stage of the sun. I don't expect it would be close to enough to become a brown dwarf star, but I think there will be some added hydrogen and helium during that time.
The heat will make it expand a bit and it will probably get darker. Losing its lighter bands which are lighter in color due to ice. But it'll still basically be Jupiter. Hotter around the surface but otherwise not very different.
Europa, however, will probably become an ocean moon instead of an ice moon, at least for a while.