What will happen in principle if one tries to push a neutron star? When we push an object it moves due to the mutual repulsion between the electrons in our hand and the electrons in that object. Since a neutron star contains only neutrons, what will happen in principle if one tries to push it?
 A: Despite pop-science descriptions, neutron stars do not contain only neutrons. They contain both protons and electrons as well, and though there are many fewer of them the electrons are also degenerate.
Accordingly it would be electron interactions (not those of neutrons) that transmitted that vast majority of the force between a normal matter pusher and a neutron star---if you could keep a normal matter pusher intact, which you couldn't.
A: Repulsion happens with any fermion particles, not just electrons. If neutrons would not repel each other we wouldn't have neutron stars. All of them would always collapse into black holes instead.
So neutrons in our atoms still repel neutron star.
However what would really happen is, neutron star has really massive gravity to the point that protons and electrons would merge into neutrons. Any kind of atom couldn't keep being atom anymore.
So when anything tries to touch neutron star, it would be suck in by gravity and collapse into lump of neutrons and feed their mass into that neutron star.
And if it collects enough mass it would collapse into a black hole.
