Mdthbs

We observe that protons are positively charged, and that neutrons are strongly attracted to them, much as we would expect of oppositely charged particles. We then describe that attraction as non-electromagnetic "strong force" attraction. Why posit an ersatz force as responsible, rather than describing neutrons as negatively charged based on their behavior?

I keep running up against circular and tautological reasoning from the laity in explanation of this (i.e. "We know they aren't charged because we attribute their attraction to a different force, and we ascribe this behavior to a different force because we know they aren't charged").

I'm looking for an empirically-based (vs. purely theoretical/mathematical) explanation.

Can someone help?

Free neutrons in flight are not deflected by electric fields. Objects which are not deflected by electric fields are electrically neutral.

The energy of the strong proton-neutron interaction varies with distance in a different way than the energy in an electrical interaction. In an interaction between two electrical charges, the potential energy varies with distance like $1/r$. In the strong interaction, the energy varies like $e^{-r/r_0}/r$, where the range parameter $r_0$ is related to the mass of the pion. This structure means that the strong interaction effectively shuts off at distances much larger than $r_0$, and explains why strongly-bound nuclei are more compact than electrically-bound atoms.

Suppose that the strong nuclear force were instead caused by Coulomb interactions. Since we know how strong the binding energies are (of the order of $sim 1text{MeV}$, as can be gleaned by say, looking at a table of alpha particle energies) and how far apart the nucleons are (about a proton radius, or $a_psim1text{fm}$) we know how charged the neutrons must be.

A quick estimate is given by letting the charge on the neutron be $-Ze$ then the binding energy is of order:

$$ frac{Ze^2}{4 pi epsilon_0 a_p} sim 1text{MeV}$$

This gives $Z sim 0.7$ which is just ludicrously large and would have been noticed in experiments of neutron paths in electric fields as noted in @rob's answer.

Which is to say: the direct experimental limit on the charge of the neutron is low enough that the electrostatic binding energy cannot account for the nuclear binding energy.

As Richard Feynman pointed out in his lectures "The Character of Physical Law", the ultimate test to decide whether or not a theory is correct is the experiment. Rob correctly stated there is strong evidence suggesting the null interaction between a neutron and some external electric influence. Measurements about masses and electric charges of several atomic components have been made with increasing accuracy, with Robert Millikan's oil drop experiment and others like it (Wilson's cloud chamber) being reasonably convincing about the "granular" nature of electric charge.

As the accuracy began to improve, it was possible to test such hypothesis as the compound nature of an atom nucleus, so that borrowing from chemistry the concept of isotope, experiments gave strength to the proposal of the neutron as a "companion" of the proton inside the nucleus. Further hypothesis made with those new considerations were experimentally proven to be correct, so there was more and more evidence to think of the neutron as a particle with no net electric charge.

There is no reason to take that fact as an axiom, however; as Einstein said once, "No amount of experimentation can ever prove me right; a single experiment can prove me wrong". Until now, the neutral behavior of the neutron has proven to be right.