What would be the characteristics of Jupiter if it shrank?
While you can't turn Jupiter into a star, it is not ruled out that you could turn Jupiter into a catastrophic thermonuclear bomb. The limitations to this was calculated at Lawrence Livermore in the 1970s, as a continuation of the work done to check to make sure Earth's oceans wouldn't ignite due to the deuterium content of water. Necessary Conditions for the Initiation and Propagation of Nuclear Detonation Waves in Plane Atmospheres by Weaver and Wood, couldn't rule out a self-sustaining ignition shock-wave in a planetary atmosphere at a deuterium concentration of more than 1 percent at ordinary liquid densities.
Although this makes the oceans safe, Jupiter is big, and it might have segregated a deuterium layer deep inside which has a high enough concentration to allow a self-sustaining nuclear ignition. Then if you drop a configuration of plutonium designed to detonate the deuterium by a nuclear explosion at the appropriate depth, you could get a detonation wave that ignites the entire deuterium layer within a very short time, the time it takes a shock wave to encircle Jupiter.
The energy output could convert a non-negligible fraction of the deuterium in Jupiter to 3He/tritium, and release enormous amount of energy. If 1 Earth mass of deuterium is ignited by the ignition shock wave, the energy release is 1038 J, over a very short time, perhaps an hour or two and this is already 10,000 times the energy output of the Sun in a full year. The resulting explosion would destroy that part of the world facing Jupiter, and probably bake the rest. I don't lose sleep over this, though.
If there is a natural trigger for such an explosion, perhaps the collision of a rocky planet with a gas giant, one might experimentally observe such planetary mini-supernovas somewhere. This was suggested in section VIII of Weaver and Wood's paper.
An object of (relatively low) Jupiter's mass cannot become a star because the minimum mass for which the density becomes high enough turn the object into star is 60 times higher.
Jupiter's actual mass is $2\times 10^{27}$ kilograms, i.e. 320 times the Earth's mass or 1/1,000 of the solar mass. For 60 times higher masses, any addition of mass would be enough to make the planet collapse, instead of grow, and fusion would ultimately get ignited.
Even if you compressed the current Jupiter's mass into a small volume so that fusion would start, it wouldn't be sustainable because the gravity wouldn't be sufficient to sustain the high density. The intense pressure from the radiation created by the fusion would make the object explode, anyway.
If your plan is to design some mechanism to mechanically keep the volume of Jupiter tiny – such a mechanism is not only infeasible for a civilization that doesn't even fully control the Earth but it may perhaps be prohibited by general principles of physics (because the huge pressure around the "confined Jupiter" could only sustained by matter in the state of fusion or even more extreme forms of matter, so effectively, the "whole" star would be much heavier than Jupiter, anyway) – that's fine but then the question becomes meaningless.
Yes, if someone is equipped with divine powers not allowed by the laws of Nature as they are currently understood, He may turn Jupiter into a star, bread into gold, and water into wine, among millions of other wonderful things. But these prospects are not a topic in physics; it's debatable whether theologians would agree that they belong at least to theology: I don't think so. Physics studies Nature as it actually works and if it finds out that something doesn't work in a certain way, it must take this insight seriously rather than to overlook it.