I started with this:
- - X X X X - - - - X X X X - - X X - - - - X X X X - - - - X X X X - - - - X X X X - - - - X X - - X X X X - - - - X X X X - -
Pushed things this way and that, ended up with this:
- - - X X X - X X - X X - - - X X X X - - - - X X - X - - - X X X X - - X - X - - X - - X X X - - X X X - X - - - - - X X X X -
Similarly, on 9x9:
- - - X X X - - X - - X X X - - - X X X X - - - - - X X X - - - - - X X X - - - - X X X - X X - - - X - X - - X X - - - X X - - - X X X - X - - - - - X X X X - -
And on 10x10:
- - X X X X - - - - - - - X X X X - - - - - - - X - X X - X X - - - - - - X X X X X - - - - - - X X X X - - - - - - X X X X - - - - - X X - - X X - - - X X - - - - X X - X X - - - - - X X X X - - - -
It took me a while to get there, but that one suggests an emerging pattern.
And here is an expandable solution for any $2n\times 2n$ grid.
