Does pair production happen even when the photon is around a neutron?

Quantum mechanics says that everything that is not forbidden is compulsory. Any process that doesn't violate a conservation law will happen, with some rate or cross-section. However, this general principle doesn't tell you what the rate is. For example, it's theoretically possible for 124Te to decay into two 62Ni nuclei plus four electrons and four antineutrinos, but to predict the (very small) rate, you need to know the relevant nuclear physics.

In your example, the process probably would go at some rate determined by electromagnetic interactions, because the neutron has a magnetic field. But the rate would presumably be small because the magnetic field of a dipole falls off like $1/r^3$, and magnetic effects are usually down by $\sim v/c$ compared to electric effects.

Yes, pair production can occur even near a lone neutron. The presence of a mass for the photon to interact with is required for conservation of momentum (further explanation can be found here). To my knowledge, the charge of the nucleus is not significant to the process of pair production, though the probability of pair production increases approximately with atomic number squared. Experimentally, this would be hard to demonstrate as neutrons are hard to control and have a relatively short half life (~10.3 minutes).

If you think of pair production as the exchange of a photon between a recoil nucleus and either the electron or the positron in the final state, then that is not going to happen with a neutron which doesn't have electric charge for the photon to couple to.

There are two ways to convince oneself that pair production off neutrons nevertheless happens. The first stays close to the picture you probably have of pair production, the second goes a bit deeper into what these pictures mean or express:

1) the neutron is not an elementary particle, it is composed of quarks which are charged. The photon that balances the recoil momentum with the mass-shell requirement can interact with those quarks instead of the neutron as a whole. It is just very unlikely: the electrical field of the neutron is limited to its interior, whereas the field of a proton (in the usual case) extends to the whole atom whose nucleus it is contained in.

2) taking a step back, and more in line with the previous answers arguing based on unitarity (i.e. "anything that can happen, happens"): what is pair production? Above we imagined a possible Feynman diagram for pair production. But that is not what nature does. We don't know what nature does. All we do is observe the following: photon and neutron go in, positron, electron and neutron go out. What happened in between, we can't know. In Quantum Field Theory we use a so-called on-shell renormalization scheme which makes the objects that we use for calculation very similar to the objects we observe in the lab, and thus allows us to describe processes with great accuracy with little calculatory effort -- but nevertheless, we are still only looking at part of could happen in the process "photon + neutron in; e-, e+, n out". So in that sense the mental picture that I gave in 1) doesn't describe what happens in nature, not even in the case where the recoil is a proton. It's just a convenient approximation.

Now given that, we can let our phantasy roam and invent all kinds of intermediate processes that would lead to the observation, while being able to balance energy and momentum (i.e. the photon mass). E.g., the photon could split into an intermediate electron and positron, and the positron could then be absorbed by the neutron (or rather its constituents, but let's ignore that, hadronization is hard), which would now now in some state of charge 1, say a proton, but a $\Delta^ +$ resonance also fits the bill. Conservation of Lepton number (a fundamental law) requires simultaneous emission of a neutrino. The electron and the neutrino meet, become a $W^-$ boson, and the $W^-$ boson could radiate off a $Z$ boson before being absorbed by the proton, turning the proton back into a neutron. Finally the Z could decay into two electrons.

This process is infinitely unlikely. But since we only observed $\gamma, n$ going in, and $e^+, e^-, n$ going out, who is going to tell nature that this isn't what she did?

The intermediate processes that we imagine are just tools to guide us through the calculation, they do not describe what nature does.