I would definitely lean towards "nothing" happening.
The "goes dark for 200 years" answer makes an awful lot of assumptions, some of which seem unfounded to me. In particular, it assumes that Jupiter will evenly spread over the surface of the sun, and will remain on top without mixing with the bulk of the sun.
At one other extreme, if Jupiter does not spread at all, it can at most blot out around $.3\%$ of the sun's luminosity (as that is the fraction it would cover). They also mention that the mass would make the sun last longer, which is suspect, since extra mass typically reduces the lifetime of main sequence stars.
Given that sunspots larger than Jupiter form occasionally, and are quickly mixed back into the sun, it seems likely Jupiter would do the same thing. And if it is mixed through the convection layer, all it would do is marginally decrease the temperature, and thus marginally dim the sun.
