  The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a
largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the
cardinality of a largest open packing of $G$, where an open packing (resp.
$2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which
intersect. It is proved that if $G$ is bipartite, then $\rho^{\rm o}(G\Box K_2)
= 2\rho_2(G)$. For hypercubes, the lower bounds $\rho_2(Q_n) \ge 2^{n - \lfloor
\log n\rfloor -1}$ and $\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor
-1}$ are established. These findings are applied to injective colorings of
hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest
hypercube which is not perfect injectively colorable. It is also proved that
$\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$, where $H$ is an arbitrary
graph with no isolated vertices.
