Yeah. Did some reading up on it. Jupiter's mass is only about 1000th of the Sun's; it would need only about 13 Jupiter masses to become a small brown dwarf, but 75 Jupiter masses to become a large brown dwarf (or small red dwarf).
That said, the sun will lose at least half it's current mass over the next 5 billion years or so. That's 500 Jupiter masses worth of material, and Jupiter is in the right place to collect some of it, being the closest gas giant. That said, the increased solar wind of the sun's red giant phase might actually end up eroding part of Jupiter's atmosphere. Still, I suppose it's theoretically possible.
Theoretically, yeah, but you think about the scale of everything, all of that material isn't going to come off the Sun, it's going to be partially the material swept away when carbon can't fuse and the hot core at the centre for its white dwarf phase.
The actual scale of the whole thing (the Sun has a radius of about 700,000km while Jupiter is at 770,000,000km away with a radius of 70,000km) just makes it impossible. All the matter will be scattered in a (roughly) spherical shape, so as well as pushing Jupiter out further, only a tiny concentration of the matter will ever get caught up in its orbit, never mind the gravity well itself.
Sorry to be a negative numpty!