  Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial
domain $\Omega^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$
be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable
spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the
Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the
product domain $\Omega^{(1)}\times\cdots\times \Omega^{(n)}$. We show that if a
function $F(\lambda_1, \ldots ,\lambda_n)$ satisfies a Marcinkiewicz-type
differential condition, then the spectral multiplier operator $m(\Box_b^{(1)},
\ldots, \Box_b^{(n)})$ is a product Calder\'on--Zygmund operator of Journ\'e
type.
