Neutrons are composite neutral particles, but their internal valence quarks are electrically charged. So why don't neutrons emit Cherenkov radiation? Is it because the distance between the quarks is too small to create such an effect?
In this definition (in blue) the phrase 'Cherenkov radiation is electromagnetic radiation emitted when a charged particle...' indicates that by this definition the neutron, an neutral particle, does not emit Cherenkov radiation.
The emission of Cherenkov radiation is itself due to an asymmetric polarisation of the medium in front and at the rear of the particle. This gives rise to a varying electric dipole momentum.
Saying this another way, when a charged particle moves through the medium at a speed higher than the speed of light in that medium, this excites the medium. The medium returns to the ground state by emitting photons of light. This is what gives Cherenkov radiation its characteristic blue glow.
This light propagates in a cone shape, with formular similar to that describing a sonic boom:
$$ \cos(\theta) = vlight/v $$
where $\theta$ is the cone's half angle, and v is the speed of the particle.
As this is on the whole particle scale, it is the net neutrality of the neutron that is important. In the definition (in blue) it is important to read 'particle' as meaning the particle that you have. In this case you have whole neutrons. They may be made up of elementary particles, but you still have neutrons as your particles (the quarks are not free to be on their own outside of the neutron).
In summary, as neutrons are as a whole neutral, they do not cause the polarisation required for Cherenkov radiation to take effect. This is the particle you have given in question of whether it emits Cherenkov radiation, so you have to work at that (composite particle) scale.
Additionally, it is sometimes stated that neutrinos 'emit' Cherenkov radiation, however this is not strictly true. It is actually resultant muons and electrons that result in the Cherenkov radiation.
Specifically, when a muon neutrino interacts with a nucleus, it can produce an energetic muon. This emits a sharply outlined cone of Cherenkov radiation which can be detected by photomultiplier tubes.
An electron neutrino interaction can produce an energetic electron, but the Cherenkov cone from this interaction differs significantly from that of the muon. The electron generates a shower of electrons and positrons, each with its own Cherenkov cone. The resultant diffuse circle at the photomultipliers is the signature of the electron neutrino. Tau neutrinos are not detected by these detectors because the neutrino energies are not sufficient to produce tau particles Neutrinos themselves do not fit the definition.
Actually, a neutron interacts with electromagnetic fields through its magnetic moment. A Cherenkof effect is therefore possible and I remember I had to calculate it as part of a home work when I was a student in physics. I lost the calculations and I do not remember the numerical results, except that the effect is much smaller than for a charged particle.