Neither of those statements are true. It's an easy approximation to make: a neutron star has all of that 'space' removed from between nucleons --- so we just need to know how big a neutron star of mass equal to the solar system would be. Well, the only significant mass is the sun (jupiter is about 1% the mass of the sun---negligible). If the sun were compressed into a neutron star, it would have a radius of about 10km (up to 50% or so accuracy). See this nice talk about neutron star radii.
Solar System:
So if you removed all of the 'space' between all of the atoms in the solar systems, it would form an object about the size of a large town, or small city.
Universe:
Obviously collecting all of this mass would yield a black-hole. But conceptually, using some very order of magnitude estimates for the universe as a whole, if we assume there are roughly $10^{20}$ - $10^{22}$ stars (I think this estimate is quite high), then the radius would be something like a 1-100 Mpc or roughly 10 million to 1 billion light-years.
Edit (To address the question itself):
The concept of 'size' for atoms and nuclei has some grey area, but you can define the size of a hydrogen atom, or the size of a proton/neutron to an order of magnitude. A statement like 'remove all of the empty space' is much more nebulous, and ends up being largely a question of semantics. A more accurate way of phrasing the underlying concept being addressed might be something like:
'Roughly how much volume do the dominant mass-constituents of matter take up?'
The idea is that nucleons (protons and/or neutrons) are 2000 times more massive than electrons, and thus the important component of mass. At the same time, the electrons are the dominant volume-fillers (by a factor of about $10^{15}$).