Jupiter is not nearly heavy enough to sustain a fusion reaction. Fusion only takes place under extreme circumstances (temperature, pressure). It happens in stars, because they are so big that their own gravity compresses the core sufficiently to create these circumstances. Detonating a bomb on/in Jupiter won't do anything to change its mass or density.
Furthermore, hydrogen fusion isn't a chain reaction process in the way that nuclear fission is. Nuclear fission (as used by humans anyway) relies on energetic neutrons striking the atoms of the fissionable material (e.g. uranium), causing it to break apart and emit more energetic neutrons, which will cause fission in other atoms, etc... a chain reaction.
Hydrogen fusion doesn't need an external input, such as neutrons, like fission does. Simply crank up the temperature and pressure of a hydrogen gas and eventually fusion will take place. This means that starting the process isn't a matter of "kickstarting" it.