The answer is yes (at least for $\kappa = \omega_1$): Shelah and Harrington showed that you can force that every stationary subset of $S_{\omega}^{\omega_2}$ reflects starting with a Mahlo cardinal. See Theorem A in Some exact equiconsistency results is set theory.

Since $\neg \square(\omega_2)$ implies that $\omega_2$ is weakly compact in $L$, if we start with a Mahlo cardinal $\kappa$ which is not weakly compact in $L$ and collapse it to $\omega_2$ using the forcing of Shelah-Harrington we'll have $\square(\omega_2)$ in the generic extension.

I don't know if this is true for $\kappa > \omega_1$ as well.

The answer is yes (at least for $\kappa = \omega_1$): Shelah and Harrington showed that you can force that every stationary subset of $S_{\omega}^{\omega_2}$ reflects starting with a Mahlo cardinal. See Theorem A in Some exact equiconsistency results is set theory.

Since $\neg \square(\omega_2)$ implies that $\omega_2$ is weakly compact in $L$, if we start with a Mahlo cardinal $\kappa$ which is not weakly compact in $L$ and collapse it to $\omega_2$ using the forcing of Shelah-Harrington we'll have $\square(\omega_2)$ in the generic extension.

I don't know if this is true for $\kappa > \omega_1$ as well.