Via integer linear programming, I found the following minimum values for $n \times n$ grids:
\begin{matrix} n & \min \\ \hline 3 & 4 \\ 4 & 6 \\ 5 & 8 \\ 6 & 11 \\ 7 & 14 \\ 8 & 16 \\ 9 & 20 \\ 10 & 24 \\ 11 & 28 \\ 12 & 32 \end{matrix}
8x8:
9x9:
10x10:
11x11:
12x12:
