  A "Higgs factory", an electron-positron collider with center-of-mass energy
of about 250 GeV, will measure the cross section of the Higgsstrahlung process,
$e^+e^-\rightarrow hZ$, with sub-percent precision. This measurement is
sensitive to a variety of new physics scenarios. In this paper, we study two
examples. First, we compute corrections to the $e^+e^-\rightarrow hZ$
differential cross section in the effective field theory (EFT) approach,
including the complete set of dimension-6 operators contributing to this
process. These results are applicable to any model where the new physics mass
scale is significantly above the weak scale. Second, we present a complete
one-loop calculation of the effect of third-generation squarks, with arbitrary
soft masses and mixing, on this cross section. This is expected to be the
leading correction in natural supersymmetric models. We demonstrate the
agreement between the full one-loop calculation and the EFT result in the limit
of large stop masses. Finally, we estimate the discovery reach of the
$e^+e^-\rightarrow hZ$ cross section measurement in the two models.
