  We define the degenerate two boundary affine Hecke-Clifford algebra
$\mathcal{H}_d$, and show it admits a well-defined $\mathfrak{q}(n)$-linear
action on the tensor space $M\otimes N\otimes V^{\otimes d}$, where $V$ is the
natural module for $\mathfrak{q}(n)$, and $M, N$ are arbitrary modules for
$\mathfrak{q}(n)$, the Lie superalgebra of Type Q. When $M$ and $N$ are
irreducible highest weight modules parameterized by a staircase partition and a
single row, respectively, this action factors through a quotient of
$\mathcal{H}_d$. We then construct explicit modules for this quotient,
$\mathcal{H}_{p,d}$, using combinatorial tools such as shifted tableaux and the
Bratteli graph. These modules belong to a family of modules which we call
calibrated. Using the relations in $\mathcal{H}_{p,d}$, we also classify a
specific class of calibrated modules. The irreducible summands of $M\otimes
N\otimes V^{\otimes d}$ coincide with the combinatorial construction, and
provide a weak version of the Schur-Weyl type duality.
