  The minimal $U(1)_X$ extension of the Standard Model (SM) is a simple and
well-motivated extension of the SM,
  which supplements the SM with the seesaw mechanism for naturally generating
the light neutrino masses
  and offers various interesting phenomenologies. In the model, the $U(1)_X$
charge of each SM field is characterized
  by the $U(1)_X$ charge of the SM Higgs doublet with a free parameter $x_H$.
Due to the $U(1)_X$ charge of the Higgs doublet, the Higgs boson has a
trilinear coupling
  with the $Z$ boson and the $U(1)_X$ gauge boson ($Z^\prime$). With this
coupling, a new process for the associated Higgs boson production
  with a $Z$ boson arises through a $Z^\prime$ boson in the $s$-channel at high
energy colliders. In this paper, we calculate the associated Higgs boson
production at high energy colliders
  and show interesting effects of the new $Z^\prime$ boson mediated process,
  which can be tested in the future.
