Do the electrons really get squeezed closer to their nuclei? I thought that only happened in massive collapsing stars where gravity overpowers everything.
There's a continuum. In the cores of stars that continuum bumps up against the physical limits of electron degenerate (white dwarf) matter. But even within ordinary planets like our own solid matter is compressed to a significant degree. For example, even though Earth's inner core is primarily comprised of Iron and Nickel it is compressed to such a degree that it is denser than Lead.
"The greater the compression, the greater the density" is essentially tautological. Of course packing the same amount of stuff in a smaller space makes it more dense. It's true in the same way the statement "Enlarging things makes them bigger" is true.
But as for gravity, no. Gravity is a function of mass. Compressing a body into a smaller space will not increase the gravity you experience at a given distance from the mass's center of gravity (though it will allow you to get closer to the center of gravity). If you magically replaced the Sun with a black hole of equal mass, the planets would continue their orbits undisturbed — even though a black hole's singularity is infinitely dense and the Sun is less dense than the Earth on average.
The general theory of relativity does a really, really good job of describing gravity as a result of mass changing the shape of spacetime, and predicting its effects as a result of that. The strength of gravity only depends on the total mass, not its density, so I'd say that gravity is not an emergent phenomenon from the density of matter and energy.
(Mass is almost certainly caused by the interaction between some quantum particles and the Higgs field. I'm not clear on how that deforms spacetime, though.)
The article begins with "in Earth's deep interior squeeze atoms and electrons so closely together that they interact very differently". The phrase is a little misleading.
The "position" of the inner electron orbitals is mostly determined by quantum rules (it is more complicated, but think that because of something like the Pauli Exclusion Principle the electrons can’t be very close together).
The conductivity of the material depends on the farther electrons, but it is not possible to bring them closer to the core because that space is "filled" with the inner orbitals. The farther orbitals of the atoms get mixed and are transformed into bands, which are not localized in an atom, but span all the crystal.
With more pressure, the nuclear cores get closer, but not very much. The main change is how the farther electron orbitals interact. So at some distance it is possible that some orbital get mixed and at other distance another orbital get mixed. So at some distance the bands are empty or full, and at another distance the bands are partially filled. And that changes the conductivity of the material.
So probabbly the correct way to begin the article is: "in Earth's deep interior squeeze atoms so closely together that their electrons (orbitals) interact very differently".
