How, exactly, does an electron scatter off of a neutron? I thought about this recently because of news articles discussing the measurements of the 'neutron skin' of large nuclei....
Is it due to the fact that both have a negative magnetic moment?
Or to repulsion from the negatively-charged down quarks?
Or to Pauli exclusion, since a neutron can be considered an electron squished in with a proton?
 A: Electrons interact with matter (i.e., other particles) by tossing photons back and forth between them. We say that the photon is the mediator of the electromagnetic force. As pointed out by Cosmas Zachos, the electron exchanges photons with the quarks that live inside the neutron, giving rise to the scattering behavior of electrons and neutrons.
The details of the interaction between electrons of extremely high energy and neutron targets were worked out at SLAC in the late 1960's. Search on deep inelastic scattering for more information on this.
A: Neutrons are composed of 3 quarks, all of which have electric charge, but which balances: one quark has a charge of $+\frac{2}{3} e$ and the other two have $-\frac{1}{3} e$. At a suitably fine scale, i.e. "close enough" to the neutron, an electron can "see" the individual quarks and thus interact electromagnetically with them.
A simple analogy exists in classical electromagnetism. In a sense, how a charged "you" interacts with an external classical distribution of charge is determined by "how it looks": if from your vantage point, you see a point charge, then you will interact like it's a point charge, even if up close it looks different, like an irregular blob - and if you get close enough you can resolve that, then the interaction with it will change likewise. So now consider a classical electric dipole - two equal and opposite point charges separated by a distance. Very, very far away, i.e. at a vast number of multiples of the separation, that dipole's two point charges will appear to coincide and cancel out - it looks neutral, and you won't feel any force.
But now get closer to the dipole - close enough you can see the two charges as separate from each other. Now, you will feel force from them.
The same goes with the neutron, except the scale is much smaller and we also have to take into account that it is a quantum, not classical, system, including the electromagnetic field.
