Yes. Moreover, there's a trivialization of the tangent bundle of $M_\alpha$ that restricts to the standard trivialization of the tangent bundle of $T^2 \times [1/4,3/4]$. By "the standard" trivialization I mean one that's invariant under the action of $T^2$. You construct it by hand -- take the invariant one and apply your gluing map $(x,0) \sim (\alpha(x),1)$. The gluing map does not preserve the trivialization on the boundary, so on one end of the boundary you need to rotate it a little, to ensure that it is preserved.
So what I'm saying is you can do it for $spin$ structures, and every $spin$ structure induces a $spin^c$ structure, giving you what you want.