“If Jupiter and Saturn would be pushed together, would they become a Brown Dwarf or a ignite a Star?”
No. Even Jupiter and Saturn combined are not heavy enough to qualify as a brown dwarf, let alone the smallest possible red dwarf star capable of hydrogen fusion. However, there is some debate about the distinction of a giant gas planet and a very low mass brown dwarf. What criteria to use? Larger brown dwarf could be capable of fusing small amounts of deuterium (like protostars do before they become real stars).
“Could a dying sun be reignited by pushing one of the gas planets into the sun?”
Actually the opposite is true. A dying sun, or rather before it becomes a dying sun extracting gas from it would prolong its life. Pushing gas planets into the sun would force it to do die an early death. It would be even more dramatic when the sun is already dead, i.e. when it has transformed into a white dwarf. Pulling planets into it would finish it off completely, meaning you sort of vaporize the corpse. As soon as the white dwarf reaches the critical mass of 1.44 solar masses (the so-called Chandrasekhar limit) it explodes spectacularly as a type Ia supernova. Nothing remains. No neutron star, no black hole.