Let $Y_n = X_n/\mathbb{Z}_2$.

If $G$ is a finite group acting freely on a manifold $M$, and $\pi : M \to M/G$ denotes the quotient map, then $\pi^* : H^*(M/G; \mathbb{Q}) \to H^*(M; \mathbb{Q})$ is injective; moreover, the image is $H^*(M; \mathbb{Q})^G$.

Involution acts on $H^2(S^2; \mathbb{Q})$ by $-1$, so by the Künneth Theorem, $\mathbb{Z}_2$ acts on $H^{2l}(X_n; \mathbb{Q})$ by $(-1)^l$. Therefore $H^{4k + 2}(Y_n; \mathbb{Q}) \cong H^{4k + 2}(X_n; \mathbb{Q})^{\mathbb{Z}_2} = 0$ and hence $H^{4k + 2}(Y_n; \mathbb{Z})$ is a torsion group. In particular, if $Y_n$ admits an almost complex structure, the odd Chern classes must be torsion.

A product of spheres is stably parallelisable, so $p_i(TX_n) = 0$. As $\pi : X_n \to Y_n$ is a covering map, we have $\pi^*TY_n \cong TX_n$, so $\pi^*p_i(TY_n) = p_i(\pi^*TY_n) = p_i(TX_n) = 0$. Therefore $p_i(TY_n) = 0 \in H^{4i}(Y_n; \mathbb{Q})$ and hence $p_i(TY_n) \in H^{4i}(Y_n; \mathbb{Z})$ is torsion.

If $Y_n$ admits an almost complex structure, then 

$$p_i(TY_n) = 2(-1)^ic_{2i}(TY_n) +\ \text{terms involving lower Chern classes}.$$ 

Together with the fact that the odd Chern classes of $Y_n$ are torsion, it follows inductively from the above formula that all of the Chern classes of $Y_n$ are torsion. In particular, $c_{2n}(Y_n) = 0$ as $H^{4n}(Y_n; \mathbb{Z}) \cong \mathbb{Z}$ is torsion-free. But this is a contradiction as 

$$\langle c_{2n}(Y_n), [Y_n]\rangle = \langle e(Y_n), [Y_n]\rangle = \chi(Y_n) = \frac{1}{2}\chi(X_n) = \frac{1}{2}2^{2n} = 2^{2n-1} \neq 0.$$

Therefore $Y_n = X_n/\mathbb{Z}_2$ does not admit an almost complex structure for any $n$.

Although this is not part of your question, it is worth noting that this implies that $\operatorname{Gr}(2, 4)^n$ does not admit an almost complex structure for any $n$ (because $\operatorname{Gr}(2, 4)^n$ is covered by $Y_n$).