Nope. Neutron stars exist in a narrow window of masses and densities and these characteristics have been confirmed in observations of them, at least the sort that we can detect (ones that are pulsars). The neutron star itself does have some structure, though, with various layers containing slightly different densities, the different forms being called "nuclear pasta". It has a gnocchi, spaghetti, lasagna, anti-spaghetti, and finally a 'swiss cheese' layer.

I'm glad that scientists get to indulge in whimsical naming once in a while. (the truth and beauty quarks got re-named top and bottom unfortunately).

The "material" of a neutron star is just that neutrons. All of the electrons in the atoms are pressed into the protons, turning them into neutrons. I forget is this is neutron or electon degeneracy pressure... But it ceases to be made of atoms at that point.