Following Brown, Maclay, "Vacuum Stress between Conducting Plates: An Image Solution"
(DOI:10.1103/PhysRev.184.1272) consider two ideally conducting parallel conducting plates separated by distance $a$ along the $z$-axis.
Simple symmetry argument allows us to obtain the possible structure of vacuum stress-energy tensor $\langle T_{\mu\nu}\rangle_0$. The plates remain invariant under rotation in $xy$-plane as well as under boosts along any direction in $xy$-plane. Therefore tensor $\langle T_{\mu\nu}\rangle_0$ must have the corresponding $SO(2,1)$ symmetry. In addition it must be traceless. The only possible structure that satisfies these requirements is
$$\langle T_{\mu\nu}\rangle_0 = \left(\frac14 g_{\mu\nu} - \hat{z}_\mu\hat{z}_\nu\right)\cdot f(z),$$
where $\hat{z}_\mu$ is space-like unit vector along the $z$-axis and $f(z)$ is unknown yet function.
Furthermore, conservation of energy-momentum means that $f(z)$ must be constant, except at the plates, so this function takes two different values $C_1$ and $C_0$ outside and inside the plates (reflection symmetry means that values of $f(z)$ on both sides outside must be equal). Considering limits $a\to 0$ and $a \to \infty$ (which both correspond to a single mirror situation) we must conclude that $C_1 = 0$.
The remaining unknown constant $C_0$ must be computed through some sort of regularization.
Brown, Maclay use explicit Green funciton construction, one can use $\zeta$-function regularization or simply compute total energy as $E= \sum_i \hbar \omega_i /2 $.
The result is (inside the plates):
$$ \langle T_{\mu\nu}\rangle_0 = \mathrm{diag}(-1,1,1,-3) \frac{\hbar c}{a^4}\frac{\pi^2}{720},$$
and zero outside. So here it is: the negative energy... and also pressure causing Casimir force.
This result should be the stress-energy tensor standing in the rhs of Einstein equations:
$$ R_{\mu\nu} - \frac12 g_{\mu\nu} R = \frac{8\pi G }{c^4} \langle T_{\mu\nu}\rangle_0 .$$
So that is how you calculate geometry: by solving Einstein equations with stress-energy tensor including $\langle T_{\mu\nu}\rangle_0 $ (and matter, which will be present).
