  We obtain the critical magnetic field required for complete destruction of
$S$-wave pairing in neutron matter, thereby setting limits on the pairing and
superfluidity of neutrons in the crust and outer core of magnetars. We find
that for fields $B \ge 10^{17}$ G the neutron fluid is non-superfluid -- if
weaker spin-1 superfluidity does not intervene -- a result with profound
consequences for the thermal, rotational, and oscillatory behavior of
magnetars. Because the dineutron is not bound in vacuum, cold dilute neutron
matter cannot exhibit a proper BCS-BEC crossover. Nevertheless, owing to the
strongly resonant behavior of the $nn$ interaction at low densities, neutron
matter shows a precursor of the BEC state, as manifested in Cooper-pair
correlation lengths {being} comparable to the interparticle distance. We make a
systematic quantitative study of this type of BCS-BEC crossover in the presence
of neutron fluid spin-polarization induced by an ultra-strong magnetic field.
We evaluate the Cooper pair wave-function, quasiparticle occupation numbers,
and quasiparticle spectra for densities and temperatures spanning the BCS-BEC
crossover region. The phase diagram of spin-polarized neutron matter is
constructed and explored at different polarizations.
