  We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm
X_n)^2$ such that the squares $X_i^2$ are minimal squares found in optimal
squareful infinite words. We apply a method developed by the second author for
studying word equations and prove that there are exactly two families of
solutions: reversed standard words and words obtained from reversed standard
words by a simple substitution scheme. A particular and remarkable consequence
is that a word $w$ is a standard word if and only if its reversal is a solution
to the word equation and $\gcd(|w|, |w|_1) = 1$. This result can be interpreted
as a yet another characterization for standard Sturmian words.
  We apply our results to the symbolic square root map $\sqrt{\cdot}$ studied
by the first author and M. A. Whiteland. We prove that if the language of a
minimal subshift $\Omega$ contains infinitely many solutions to the word
equation, then either $\Omega$ is Sturmian and $\sqrt{\cdot}$-invariant or
$\Omega$ is a so-called SL-subshift and not $\sqrt{\cdot}$-invariant. This
result is progress towards proving the conjecture that a minimal and
$\sqrt{\cdot}$-invariant subshift is necessarily Sturmian.
