And the juicier tl;dr bits (note that XKCD only dares consider an electron moon, not a whole sun):

The amount of energy in our electron Moon, it turns out, is about equal to the total mass and energy of the entire visible universe.

[…], the energy from all those electrons pushing on each other is so large that the gravitational pull wins, and our singularity would form a normal black hole. At least, “normal” in some sense; it would be a black hole as massive as the observable universe.

Would this black hole cause the universe to collapse? Hard to say.