When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality constraint. Are there problem instances, where it makes better sense to square the equality constraint and then use that in the unconstrained problem?
When the equality constraint has the form $g(x)=0$ then replacing it with $h(x):=g^2(x)=0$ is fatal. Lagrange's method detects the points where $df(x)=\lambda dg(x)$ with a finite $\lambda$, and together with $g(x)=0$ you can expect to obtain a finite number of "conditionally stationary points" on the surface $S: \>g(x)=0$. But $dh(x)=0$ at all points of $S$, so the equation $df(x)=\lambda dh(x)$ will not produce a single point that is of interest.
It is another matter with the objective function $f$. When $f$ appears as a square root (e.g., if $f$ represents a distance between certain points) then no harm is done when you replace $f$ by $f^2$.