  We study the phenomenology of the light and heavy Higgs boson production and
decay in the context of a $U(1)_{B-L}$ extension of the Standard Model with an
additional $Z'$ boson at future $e^+e^-$ linear colliders with center-of-mass
energies of $\sqrt{s}=500-3000\hspace{0.8mm}GeV$ and integrated luminosities of
${\cal L}=500-2000\hspace{0.8mm}fb^{-1}$. The study includes the processes
$e^{+}e^{-}\rightarrow (Z, Z') \to Zh$ and $e^{+}e^{-}\rightarrow (Z, Z') \to
ZH$, considering both the resonant and non-resonant effects. We find that the
total number of expected $Zh$ and $ZH$ events can reach 909,124 and 97,487,
respectively, which is a very optimistic scenario and thus it would be possible
to perform precision measurements for both Higgs bosons $h$ and $H$, as well as
for the $Z'$ boson in future high-energy and high-luminosity $e^+e^-$ colliders
experiments. Our study complements other studies on the B-L model and on the
Higgs-strahlung processes $e^{+}e^{-}\rightarrow (Z, Z') \to Zh$ and
$e^{+}e^{-}\rightarrow (Z, Z') \to ZH$.
