The notation $\prod_{g\in G} g$ for a finite group not well-defined.

One might interpret the statement as saying that the product of all elements taken in any order gives the unique element of order $2$. But this fails for the quaternion group, which has $-1$ as unique element of order $2$, while the product $1ijk(-1)(-i)(-j)(-k)$ equals $1$, not $-1$.

The statement is wrong, not well-defined, and in fact one should assume that $G$ is abelian. Then the proof is easy by using the equivalence relation $x \sim y \Leftrightarrow x=y \vee x=y^{-1}$.

Indeed, the group needs to be commutative for this result to hold. It might be nice for someone to post a minimal counterexample! (Added: never mind: Marc van Leeuwen's answer and the comments below it take care of this.)

Because of a request from a colleague, a little while back I had the occasion to write down an excruciatingly elementary proof of this fact. See here.