Magnitude response of mirrored (with respect to unit circle) poles and zeros
You're right, the contribution of a zero $z_0=re^{j\phi}$ to the squared magnitude response is
$$\big|e^{j\omega}-re^{j\phi}\big|^2=1-2r\cos(\omega-\phi)+r^2\tag{1}$$
From $(1)$ it is clear that another zero resulting in the same frequency dependence as in $(1)$ must have the same phase angle $\phi$. Assuming $z_1=Re^{j\phi}$ we get
$$\big|e^{j\omega}-Re^{j\phi}\big|^2=1-2R\cos(\omega-\phi)+R^2=R^2\left(1-\frac{2}{R}\cos(\omega-\phi)+\frac{1}{R^2}\right)\tag{2}$$
Comparing $(1)$ and $(2)$ we get $R=1/r$ and a squared magnitude scaling of $R^2=1/r^2$. Consequently, the other zero $z_1$ is given by
$$z_1=\frac{e^{j\phi}}{r}=\frac{1}{z_0^*}\tag{3}$$
and
$$\big|e^{j\omega}-z_0\big|=r\left|e^{j\omega}-\frac{1}{z_0^*}\right|\tag{4}$$