Equivalance Operator - Perfect Secrecy
Actually, for any group operator $\odot$, we see that $x \odot y$ gives perfect security, assuming that $y$ is uniformly random over all possible group members.

The $\equiv$ operation is a group operation, with the group members being $\{0, 1\}$, with $1$ being the identity.  Hence, yes, it gives perfect security.