Well, being a terrestrial planet Mars is a lot denser than your typical gas giant. 3.9 vs 1.3 for Jupiter, and (famously) 0.687 for Saturn.
Given such a low average density, you'd have to sink pretty far into Jupiter's atmosphere before you'd float. And that's assuming your own density doesn't increase as well, which it almost certainly would.
A while ago I got curious about whether a gas balloon (as opposed to a hot-air balloon) could work on Jupiter. Although most of its atmosphere is hydrogen, a substantial fraction (24%) of its mass is helium, so a balloon containing pure hydrogen could still float. I found that for reasonable balloons and payloads you'd have to operate well below the cloud decks at the 5-6 bar level (IIRC), and that wouldn't be very scientifically interesting. It would also be pretty dark.
The hot air balloons that have been proposed could operate above the clouds in the <1 bar region.