If in a group G an element $x$ has the property such that $x^2 = e$, does that entail $x = e$? I can't seem to prove that $x = e$ from $x = x^{-1}$.

That's because it can't be done. Consider the group consisting of $-1$ and $1$ under ordinary multiplication.

There are many other examples. An element $x$ such that $x^2=e$ but $x\
e e$ is called an element of order $2$. Many groups have one or more elements of order $2$.

Consider for example the group of distance-preserving mappings from the plane to itself. Let $a$ be rotation about the origin through $180^\circ$. Then $a^2$ is the identity, but $a$ is not. Let $b$ be reflection in a certain line $\ell$. Then $b^2$ is the identity, but $b$ is not.

Or else consider the group of all permutations of the set $\\{a,b,c,d,e\\}$. Let $\sigma$ be the permutation that interchanges $a$ and $b$, and leaves the others alone. Then $\sigma^2$ is the identity, but $\sigma$ is not.