What is a neutron's polarizability?

The "crude model" that you link to is, essentially, dimensional analysis. It states that the polarizability $alpha$ should more-or-less obey $$ fracalpha{4piepsilon_0} approx a^3 $$ where $a$, which must have units of length, is the only length scale available in the problem: the radius-or-diameter of the atom's charge distribution. For hydrogen, we have the "Bohr volume" $frac{4pi}3 a_0^3 approx frac12rm,Å^3$ and, experimentally, $alpha/4piepsilon_0 approx frac23rm,Å^3$. Larger atoms tend to be more polarizable than smaller atoms, as well.

If we guessed that the similarly-normalized polarizability for the neutron were comparable to the neutron's volume, we'd be in for a bit of a shock: the neutron's volume is $frac{4pi}3 rm(1,fm)^3$ but its polarizability has been measured to be about a factor of a thousand smaller, $10^{-3}rm,fm^3$.

But a neutron doesn't look very much like a neutral atom, either. An atom has a pointlike positive charge at the center and most of its (central) volume has roughly uniform negative charge. The neutron seems to have a negative core, a positive skin, and a negative halo. Perhaps this multi-component charge distribution allows a sort of Schiff screening to reduce the apparent dipole effect? Or perhaps, if your model for distribution of charge within the neutron is that it spends some fraction of its time as a virtual proton or delta orbited by a virtual pion, the strong interaction between those components is so "stiff" that the electric field just doesn't disturb it very much?

The reference you found in your question is a 2009 report on a lattice QCD computation which gets the neutron polarizability wrong by a factor of three. If that was the state of the art eight years ago, it's probably fair to complain that "crude calculations" of the neutron polarizability are the best that anybody can do right now.