  We introduce equivariant Chow-Witt groups in order to define Chow-Witt groups
of quotient stacks. We compute the Chow-Witt ring of the moduli stack of stable
(resp. smooth) elliptic curves, providing a geometric interpretation of the new
generators. Along the way, we also determine the Chow-Witt ring of the
classifying stack of $\mu_{2n}$.
