I can't comment on the question, so I will suggest an approach here. Resize the targets (partial checkerboards in R^2) by dividing by n, and stop when the union of the resizings covers the plane (or enough of it). Hint: don't expect a maximum for n.
Now I notice n is an arbitrary real, and not an integer. I believe there is an upper bound for n, simply because the squares block all lines of sight from the origin. 3 is looking reasonable as a bound now.
Now that I have drawn a picture of squares in a plane, it is evident to me that a number near 21/5 is the bound, as any ray from the origin through a point with coordinates greater than 1 must intersect a square at the next largest permissible coordinate in the "slower" direction which will happen before n is 5 and usually much sooner. I think a pair near (1, 5/3) will be close to the extreme case.
I see I have taken the wrong squares (-1 to +1 mod 4). The analysis to the posted problem is similar, but the constants change. I leave the details to others.