Why can’t you square both sides of an equation?

Why can’t you square both sides of an equation?

I’ve been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just searched the web and still couldn’t find a simple-to-understand answer for why squaring both sides gives you extraneous solutions.

If two things are equal, then so long as you do the same thing to both, they will remain equal. There is nothing wrong with taking the square of both sides of an equation. However, you have to be careful if you want to take the square root of both sides, because the square root is not a normal function: it has two values $\pm \sqrt x$. By convention, the positive square root is chosen, and that is what people mean when they say “the square root”. But equations don’t care about our conventions. The fact that $(-1)^2 = 1^2$ certainly doesn’t imply that $-1 = 1$.
In other words, if $x^2 = y^2$, then taking the square root (using the stated convention) of both sides results in $|x| = |y|$, not in $x=y$.
For these reasons, if you have an equation containing an unknown, then squaring both sides of it can introduce new solutions, so you have to be careful. For instance, the equation $x=1$ obviously has only one solution (namely $x=1$!) but squaring both sides of it yields the equation $x^2=1$ which has the two solutions $x=\pm 1$.