Standard words and solutions of the word equation $X_1^2\cdots X_n^2=(X_1\cdots X_n{)}^2$

We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$suchthatthesquares$X_i^2$ 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 $\gcd (\mathrm{\mid }w\mathrm{\mid },\mathrm{\mid }w{\mathrm{\mid }}_1)=1$. 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 $\mathrm{\Omega }$ contains infinitely many solutions to the word equation, then either $\mathrm{\Omega }$ is Sturmian and $\sqrt{\cdot }$-invariant or $\mathrm{\Omega }$ 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.