Information on More Neighbours

Red shaded, green not shaded. This is the only configuration that works

Let's figure out what we can.
Start at the bottom right. Label each square $a,b$ where $a$ is the column and $b$ is the row. Therefore $3,4$ (i.e the 1) must be shaded because out of $3,4$ ; $3,3$ ; $4,3$ two are shaded - they can't be $3,3$ and $4,3$ . Also since one of $3,3$ and $4,3$ are shaded, $2,3$ ; $2,4$ ; and $4,4$ are not shaded. Therefore $1,3$ is shaded (because of $1,4$ ) so $3,3$ is unshaded ( $2,4$ ) which means $4,3$ is shaded ( $3,4$ ).

Now the top half is symmetric. $1,1$ is shaded because out of $2,1$ ; $2,2$ ; $1,1$ two are shaded. Similarly $4,1$ is shaded.

I think the best way to finish this would be guess and check as there are only two options, given any square - it is shaded or it is not shaded. Suppose $2,1$ is shaded. Then $1,2$ ; $3,1$ ; $3,2$ are unshaded - contradicting $4,2$ . Therefore $2,1$ is unshaded so $2,2$ is shaded and the rest quickly follows.