You could in principle start a nuclear fusion reaction if it had the right amounts of deuterium, even just a deuterium rich layer - and then that would probably blow off a lot of its atmosphere. Turns out it hasn't quite got enough deuterium to do that in its atmosphere though at one point there was speculation about whether it was theoretically possible to start a deuterium fusion explosion in the Jupiter atmosphere. (Physicists asked if dropping a plutonium bomb into Jupiter could cause a layer of its atmosphere to detonate if you exploded it at the right level, after calculation and study of Jupiter the eventual answer was no). It's not an absurd idea but Jupiter apparently can't do it.
But anyway even if it did have a deuterium rich layer - that's a one off very big explosion and then it's done. It's not a star, it's just a massive fusion bomb.
To make it into a star you need to add enough matter to make it a red dwarf. That means, 75 times the mass of Jupiter Jupiter mass Otherwise the hydrogen can't sustain fusion though deuterium fusion can happen in a sustained way with 13 times mass of Jupiter.
Callisto is 1,168,000 miles from Jupiter which is 0.0126 au. So even if you could add enough matter to make Jupiter into a red dwarf, they would not orbit in its habitable zone. Though - given that you are adding so much matter to Jupiter in the first place you could easily move them or just create a new Earth mass planet in an appropriate orbit.
Arthur C. Clarke's idea is pure science fiction and the maths doesn't add up.
Science fiction writers often fudge numbers to get their ideas to work, in interest of compelling fiction, e.g. Andy Weir in the Martian knowing that Martian dust storms are so feeble they can barely disturb an autumn leaf, but putting it in for reasons of narrative convenience. I don't know if Arthur C. Clarke's idea is like that, or if he thought it was in some way possible.