We study possible new physics interactions in the $ZZH$ vertex contributing to the Higgsstrahlung process $e^+ e^- \to ZH$ at proposed future $e^+e^-$ colliders using the polarization of the $Z$ as a probe. We calculate the spin density matrix of the $Z$ for the process and determine the eight independent polarization parameters of the $Z$ boson which have the potential to constrain the anomalous couplings. We study angular asymmetries using the decay leptons from the $Z$ boson which are simply related to the polarization observables. We also estimate the limits that can be placed on the anomalous couplings using measurements of these angular asymmetries at centre of mass energies of 250 GeV and 500 GeV and various combinations of polarized $e^+$ and $e^-$ beams.

19 Pages, 9 Tables, added discussion and references; matches published version