First, just to get this out of the way, a neutron has no charge, and could not be directly detected using an electromagnetic disturbance (since it does not produce an EM disturbance).
Next, a particle with a "real" mass cannot travel faster than the speed of light. To briefly describe why, I present the full form of the popular Einstein mass energy equivalence principle (which is typically notated as $E = m c^2$):
$$ E^2 = (mc^2)^2 + (pc)^2$$
In the case of a particle with sufficiently low momentum, the second term on the right vanishes, and the equation reduces to its famous form. However, a superluminal velocity requires $p>E/c$, which is only possible if $(mc^2)^2 < 0$. Since $c$ is a positive, fundamental constant of the universe (unchanging in this argument), this requires that $m^2<0$, which implies that the mass is imaginary.
There are, however, hypothetical particles (since they have never been observed to exist) called Tachyons, which have an imaginary mass. Note also that, just as a particle with a real mass is limited to travel below the speed of light, so too is a Tachyon restricted to speeds greater than $c$.
Now, to answer you question: Tachyons, should they exist, will otherwise behave like other particles, such as neutrosn, in the way they interact with our more traditional types of matter. In this way, a tachyon can sill scatter off of ordinary particles, and can, in theory, be detected with ordinary neutron detectors (something which I won't delve into here). However, as a final note, the momentum conservation gets a bit strangle, since the mass is no longer real, so a bit of reformulation of some of the relevant equations is necessary, but the principles are all essentially the same.
