I understood that everything was made of Atoms.


  An atom is the smallest constituent unit of ordinary matter that has
  the properties of a chemical element. Every solid, liquid, gas, and
  plasma is composed of neutral or ionized atoms


However with Neutron Stars;


  the basic models for these objects imply that neutron stars
  are composed almost entirely of neutrons, which are subatomic
  particles


Does this mean neutrons can exist outside of an atom, and that a Neutron star isn’t comprised of an element that is recognisable on the periodic table?

Yes neutrons can exist outside the atom (or nucleus). In free space a neutron will beta decay into a proton, and electron and an anti-neutrino on a timescale of 10 minutes. However, in the dense interiors of a neutron star, the electrons form a degenerate gas, with all possible energy levels filled up to something called the Fermi energy. 

Once the Fermi energy of the electrons exceeds the maximum energy of any possible beta-decay electron, then beta decay is blocked and free neutrons become stable. This is what happens inside a neutron star and you end up with mostly neutrons with a small fraction perhaps a few per cent electrons and protons.

In the outer parts of the neutron star, the protons and neutrons can still arrange themselves into nuclei (but not atoms), but these nuclei are extremely neutron-rich (they would not normally exist in nature) and are only stabilised against beta decay by the process I described above. The very outer envelope may consist of completely ionised iron-peak element nuclei and there may be an ultrathin (few cm) layer of recognisable ionised hydrogen, helium and carbon (e.g. Wynn & Heinke 2009).

Once the density reaches about $3 \times 10^{16}$ kg/m$^3$ it becomes more favourable for the neutrons and protons to organise themselves into "macro-nuclei" - long strings and sheets of nuclear material, known colloquially as nuclear pasta.

At higher densities still, the pasta dissolves into a soup of mostly neutrons with about 1 per cent protons and electrons.

The diagram below (from Watanabe et al. 2012) shows roughly how these layers are arranged. It should be stressed that this is based on theoretical modelling, with the theory becoming less certain the further into the neutron star you go. Testing these ideas involves nuclear and particle experiments, observations of pulsars, of neutron star cooling, of X-ray bursts, mass and radius estimates in binary systems, pulsar glitches, etc., etc. None of the details have been observationally confirmed beyond dispute, but the basic picture below fits what we know. In particular, the the crust and the n,p,e fluid regions are well understood in theory. The details of the nuclear pasta phases are still the subject of a lot of theoretical work, as are the details of superfluidity in the interior, and what happens in the very central regions (solid neutron core, extra hadronic phases, boson condensation, quark matter) is still theoretically difficult and observationally untested except perhaps to say that the softest equations of state have been ruled out by the existence of $2M_{\odot}$ neutron stars.


