It’s possible the problem is only the battery, but based on what you describe—especially the battery compartment getting extremely hot—the camera itself may also have been damaged.

From the symptoms alone, no one can tell exactly what failed. The simplest next step is to test the camera with a known-good OEM battery or the correct AC adapter, ideally at a reputable camera shop if you don’t want to buy parts first. If the camera powers up normally, the generic battery was likely the main issue. If it still won’t turn on, the camera probably needs repair.

Unfortunately, camera repair can be expensive relative to the value of an older model like the FZ20, so it may not be economical unless the fix is minor. Stop using the hot battery immediately, since overheating suggests a faulty battery or charging problem and could be unsafe.