  We present an evaluation of the $\pi^\pm$ and $K^\pm$ box contributions to
the hadronic light-by-light piece of the muon's anomalous magnetic moment,
$a_\mu$. The calculation of the corresponding electromagnetic form factors
(EFFs) is performed within a Dyson-Schwinger equations (DSE) approach to
quantum chromodynamics. These form factors are calculated in the so-called
rainbow-ladder (RL) truncation, following two different evaluation methods and,
subsequently, in a further improved approximation scheme which incorporates
meson cloud effects. The results are mutually consistent, indicating that in
the domain of relevance for $a_\mu$ the obtained EFFs are practically
equivalent. Our analysis yields the combined estimates of
$a_\mu^{\pi^\pm-box}=-(15.6\pm 0.2)\times 10^{-11}$ and
$a_\mu^{K^\pm-\text{box}}=-(0.48\pm 0.02)\times 10^{-11}$, in full agreement
with results previously obtained within the DSE formalism and other
contemporary estimates.
