Can a dying, swelling, star be rejuvinated by enveloping orbiting gas giants? Many hot jupiter type exoplanets have been found, orbiting near their parent stars.

At the end of a stars life, they swell up and sometimes envelop closely orbiting bodies.

Combine these effects and what could/would happen? Could the unfused hydrogen of a gas giant be dragged into the stars core and end up rejuvinating the star and extend its lifetime? Have these conditions been contemplated before? What might be expected to happen? Or is the environment of a stars outer layers just not dense enough to keep a gas giant from just continuing in its orbit?
Stars turn into Red Giants not because they're running out of fuel, but because they're accumulating material they can't use for fusion (yet) in the core. The star isn't so much dying of starvation as it is wallowing in its own muck.

Red giants form when the fusion is no longer taking place in the centre of the star, but instead in a shell around the centre. The centre is filled up with the products of fusion, which themselves cannot fuse until temperatures and pressures get high enough. The exact reason why this results in red giant formation eludes me still, but you'll find plenty of questions asking about it on this SE.

Furthermore, most stars that are larger than red dwarfs never fully use up their fuel. There is often plenty of hydrogen in the outer reaches of the star, way more than you'll find in Jupiter (which, as zeta-band mentions, is only about 1/1000th the mass of a sun-mass star).