Bulk nuclear matter is found in neutron stars at depths below around 1 km. Proton/neutron ratio in this matter is low (around 0.1), but is not zero. Neutronium does not exist in neutron stars (or black holes, either; for that matter).
Groups of 2 and 4 neutrons, and maybe more, have been observed. None have a half-life exceeding 10-18 sec, which means they cannot be atomic nuclei. Larger clusters will exist for even shorter times. Only the single free neutron can exist for a long time - and it cannot bind electrons, so it can't form atoms.
Theoretically, antineutrons do also exist. While this is very unlikely, if antimatter happened to form neutronium, then it would be considered element -0. Two isotopes, 1 and 2, have been confirmed. A third isotope, 4, is still being disputed.
Neutrons cannot fit into the periodic table, considering they are not atoms, and it does not have any properties that are even comparable to the other elements. However, they are showcased in nuclide charts.