Is there any way to attract or repel neutrons? I know that neutrons are neutral and have no charge, but is there any other way to interact with them?
 A: Short answer: steering neutrons is possible, but hard, and is very different from attracting or repelling charged particles.
The neutron participates in all four of the fundamental interactions: it feels the strong and weak nuclear forces; its magnetic moment couples its motion to electromagnetic fields; and like all objects embedded in spacetime, it's sensitive to gravity.
By an overwhelming margin, the most important of these is the strong nuclear force.  That's a contact force: the long-range part of the strong force is mediated by the exchange of pi mesons, which have a mass of about 140 MeV, so the associated potential $V_\text{strong} \propto r^{-1} e^{ -m_\pi r}$ peters out exponentially over a length scale of $r_\text{strong} =\hbar c/m_\pi c^2 \approx 1\rm\,fm$.  Unless a neutron's wavefunction actually overlaps with a nucleus, that nucleus is invisible and the strong force doesn't contribute anything.  Changes in neutron kinetic energy due to strong interactions, such as neutron emission from a nucleus, are typically on the scale of millions of electron-volts (that is, MeV). Hold on to that number for later.
The weak force you can treat the same way, except that the mediating bosons --- the $W$ and $Z$ --- are much heavier, so the range of the force is shorter, roughly $10^{-3}\rm\,fm$.  Any effect on a neutron's trajectory due to the weak interaction is therefore always overwhelmed by the effect due to the strong interaction. (Demonstrating that the weak interaction contributes at all requires looking for parity-violating effects, which is something that I spend a lot of time on but is tangential to your question.) I'm ignoring neutron decay here, since there's not a neutron left afterwards for you to attract or repel.
The neutron doesn't have an electric charge and therefore doesn't feel electric fields.  The neutron does have a magnetic moment and therefore does feel magnetic fields.  However the neutron's magnetic moment is quite small, $\mu \approx 50\rm\,neV/T$.  If I have some typical mega-eV neutron that's in a tesla-scale magnetic field, and the orientation of the neutron's spin relative to the field flips around, then the nano-eV change in the magnetic energy $U=-\vec\mu\cdot\vec B$ is a sub-part-per-trillion correction to the total energy. Apart from the special case below, you can't steer neutrons using magnetic fields in any practical way.
Likewise gravity: the gravitational force on a neutron near Earth's surface is $m_\text{n}g \approx 100\rm\,neV/m$.
This energy scale of $\sim100\rm\,neV$ turns up in one other place: it's a typical value for the Fermi pseudopotential, a model which becomes useful when the neutron's wavelength is long compared to the distance between nuclei in a material.  The coincidence of the magnetic moment, gravitational mass, and pseudopotential make it possible to completely trap so-called "ultra-cold" neutrons (UCNs), whose kinetic energy is lower than about 100 nano-eV. When such a neutron moves from vacuum to some material, it sees a step-function potential which is larger than its kinetic energy and gets reflected, just like you studied in introductory quantum mechanics. Thanks to the gravitational interaction, ultra-cold neutrons only bounce about a meter in Earth's gravitational field. (A recent experiment called the "Gravitrap" actually stored UCNs in a bucket without any top, and counted the neutrons by tilting the bucket to "pour" them onto a detector.) And if you store UCNs in a strong but varying magnetic field, the two spin states become "strong-field seekers" and "weak-field seekers" and spend their time in different parts of your magnet.
Ultra-cold neutrons are kind of magical, but the Fermi pseudopotential is the operating principle behind the most important practical way of steering/repelling neutrons: neutron mirrors.
If I have a neutron which is traveling nearly parallel to a very smooth surface, I can boost into a reference frame where the neutron's motion is normal to the surface and ask whether that "perpendicular kinetic energy" is larger or smaller than the pseudopotential.  If it's smaller --- that is, if the angle between the neutron's momentum and the surface is shallow enough --- the neutron undergoes "total external reflection" from the mirror.
If you assemble several such mirrors into a tube (usually four, with rectangular cross-section) you have a "neutron guide," which lets you transport a beam of neutrons many meters away from their source without much loss.  The state of the art for neutron guides is a divergence of about half a degree for a beam of thermal neutrons.
So: neutrons can be repelled by walls, are feebly attracted to Earth by gravity, and are feebly steered by magnetic field gradients, but that's pretty much it.
