The areas for the oblique squares have a simple expression in terms of a "bounding square" which is $m \times m$ and an "offset" $k.$ The bounding square of a particular oblique square is determined by drawing two vertical lines through the leftmost and rightmost vertices of the oblique square, and two horizontal lines through the lowest and highest vertices of the oblique square. The offset $k$ is the height of the leftmost vertex of the oblique square above the lower horizontal line making the bounding box. Note that we can include also the non-oblique squares by taking the offset as zero.
To see this as a diagram place the lower left corner of the bounding box at the origin. Then using offset $k$ the four points of the (typically) oblique square are
$$ (0,k),\ \ (k,m),\ \ (m,m-k),\ \ (m-k,0). $$
Then the area of this oblique square is $k^2+(m-k)^2.$ there are exactly $(n-m+1)^2$ bounding squares which are $m \times m.$ Using known formulas for sums of powers and some algebra, and also the total number of oblique and non-oblique squares in the large $n \times n$ grid, the average area of the two types of squares in the grid, oblique and non-oblique, can be expressed as
$$ \frac{2n^2+4n+9}{15}.$$ Note that in my setup there are in fact $n+1$ lattice points on each side of the large square. This is to make the actual dimensions $n \times n.$
