  The Standard-Model (SM) prediction for the CP-violating quantity
$\epsilon_K^{\prime}/\epsilon_K$ deviates from its measured value by 2.8
$\sigma$. It has been shown that this tension can be resolved within the
Minimal Supersymmetric Standard Model (MSSM) through gluino-squark box
diagrams, even if squarks and gluinos are much heavier than 1 TeV. The rare
decays $K_L \to \pi^0\nu\bar{\nu}$ and $K^+ \to \pi^+\nu\bar{\nu}$ are
similarly sensitive to very high mass scales and the first one also measures CP
violation. In this article, we analyze the correlations between
$\epsilon^{\prime}_K/\epsilon_K$ and $B(K_L \to \pi^0\nu\bar{\nu})$ and $B(K^+
\to \pi^+\nu\bar{\nu})$ within the MSSM aiming at an explanation of
$\epsilon_K^{\prime}/\epsilon_K$ via gluino-squark box diagrams. The dominant
MSSM contribution to the $K \to \pi\nu\bar{\nu}$ branching fractions stems from
box diagrams with squarks, sleptons, charginos, and neutralinos, and the
pattern of the correlations is different from the widely studied $Z$-penguin
scenarios. This is interesting in light of future precision measurements by
KOTO and NA62 at J-PARC and CERN, respectively. We find $B(K_L \to
\pi^0\nu\bar{\nu})/B^{SM} (K_L \to \pi^0\nu\bar{\nu})\lesssim 2\,(1.2)$ and
$B(K^+ \to \pi^+\nu\bar{\nu})/B^{SM}(K^+ \to \pi^+\nu\bar{\nu}) \lesssim
1.4\,(1.1)$, if all squark masses are above 1.5 TeV, gaugino masses obey GUT
relations, and if one allows for a fine-tuning at the $1\%\,(10\%)$ level for
the gluino mass. Larger values are possible for a tuned CP violating phase.
Furthermore, the sign of the MSSM contribution to $\epsilon_K^{\prime}$ imposes
a strict correlation between $B(K_L \to \pi^0\nu\bar{\nu})$ and the hierarchy
between the masses $m_{\bar{U}}$, $m_{\bar{D}}$ of the right-handed up-squark
and down-squark: sgn$[B(K_L \to \pi^0\nu\bar{\nu})-B^{SM} (K_L \to
\pi^0\nu\bar{\nu})] = $sgn$(m_{\bar{U}}-m_{\bar{D}}) $.
