If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are iids

If $X_1,\ldots,X_n$ are independent then so are $X_1^2,\ldots,X_n^2$ . To see that, take $a_1\leq b_1,\ldots,a_n\leq b_n$ and let $g:x\mapsto x^2$ . We have $P\left(\bigcap_{j=1}^nX_j^2\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^ng(X_j)\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right),$
and since $X_1,\ldots,X_n$ are independent, we have $P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right)=\prod_{j=1}^nP\left(X_j\in g^{-1}(\left[a_j,b_j\right])\right)$ and we are done.
We conclude that your formula is true if $Var(X_1^2)$ exists (but the problem doesn't come from the independence).