Neutrons interact with each other only via exchange interaction, while "every-day particles" and their temperatures are governed by electrostatic forces. What are the implications of this difference on the temperature of a collection of neutrons? In other words, what is the meaning of thermal vibration in the absence of electrostatic forces?
Neutrons interact with each other only via exchange interaction
Neutrons interact via the strong and weak forces. At low energies the interaction is principally via the nuclear force, or residual strong force, which derives ultimately from the strong force interactions between quarks. This can be described as an effective force due to the exchange of virtual pions.
The nuclear force allows neutrons to exchange momentum with each other and thereby thermally equilibrate.
The zero-range nuclear interactions felt by neutrons makes free neutrons, outside a nucleus or a neutron star, an excellent implementation of an ideal gas. However a neutron gas is a little unusual since neutron gases mostly are at such low density and pressure that the neutron-neutron interaction is very unlikely to occur before the neutron either decays or escapes the container. A neutron gas will therefore approach thermal equilibrium with the medium in the container, or the walls of the container; however a neutron gas is typically still pretty far from equilibrium and the velocity distribution is non-Maxwellian.
For example, neutrons in a reactor deposit most of their kinetic energy in a "moderator", which is often water; this is because room-temperature neutrons are much more likely to induce another fission than the "fast" neutrons which the fission actually produces. Neutron source facilities usually have "cold" moderators made of liquid hydrogen, which produce slower neutrons.
So-called ultra-cold neutrons (UCN) are actually quite stable in liquid helium. This is because the interactions between the neutrons and phonons in the helium is quantum-mechanical and the neutron-phonon scattering cross-section is small. UCN with thermal energy $\rm 100\,neV \approx 1.2\,millikelvin / \it k$ are stable in a 100 millikelvin helium bath for times roughly equal the the neutron lifetime.
Well essentially there is no much difference, at least if we are talking of temperature and we mean "average kinetic energy" as we generally do. Why? Because neutrons as "every-day particles" interact via the nucleon-nucleon potential which similarly to the one between molecules, it is repulsive at short distance and atractive at longer distances. Of course in this case the scales are very different.
You can think of nuclei as "bags of neutrons" in some way. But here, protons which repulse each other, only behave slightly different than neutrons, because the nuclear force is so strong. In fact some nuclear models describe excited nuclei as a gas of nucleons, which exhibits evaporation in a similar way as a hot droplet.