In trying to remember an example about quotient stacks, I think I've got something turned around. I am trying to determine whether the diagonal is a closed substack of the (product of) quotient stack(s) $[\mathbb{A}^1/\mu_2] \times [\mathbb{A}^1/\mu_2]$.
If we consider $\mathbb{A}^1/\mu_2$ as a variety, then the quotient $\mathbb{A}^1/\mu_2 \cong \text{Spec }k[x^2]$ as the ring of invariants, and $\text{Spec }k[x^2] \cong \text{Spec }k[x]$. So in some sense, we've forgotten the quotient and varieties are too "coarse" here to remember the quotient information.
If we consider $[\mathbb{A}^1/\mu_2]$ as a quotient stack, then there's a copy of $B\mu_2 = \{ * \} / \mu_2$ at the origin, i.e., the presence of stabilizers. I believe this implies that the affine pre-quotient $V(y^2-x^2)=V(x-y)\cup V(x+y)$ no longer corresponds to the correct diagonal object in $\mathcal{X} \times \mathcal{X}$, since the origin has too many Hom's to itself, and therefore the diagonal inside of the product is not a closed substack. Can anyone please clarify? Thanks