I think to usefully discuss this, we need to have at least a vague idea of how the Death Star works. Not the specific physics of it, obviously, but what effect it is specifically having on the planet. A planet like the Earth is not a balloon with a box full of dynamite at the centre. Even if you shot it with some hypothetical giant space laser that could penetrate to the core, there's nothing in the middle that would suddenly ignite and blow the planet into pieces. You'd kick off a massive series of volcanic disturbances, but no big bang.
When we've figured out what causes a big ball of rock and metal to break up into millions of asteroids, then we could talk about a gas giant. Someone suggested it creates reverse gravity at the centre of the planet. That would work on a gas giant as well as a rocky planet. Better in fact. But the sheer scale of gas giants would mean orders of magnitude more power. For example, you've all seen photos of Jupiter with the "red spot". That spot is in fact a giant cyclone. And you could fit two Earths side by side in it - that's how large it is.
I would say gas giants are not an option for either Death Star, personally.