Denoting the level set function by L(x,y), you could use LSQNONLIN (if you have it) to minimize L^2(x,y) subject to the constraint that x,y lie in a particular square. If the minimum is achieved at an L^2 value close to zero, you know it intersects that square.
To find the intersection points, you can then minimize L^2 subject to the constraint that it lie a line segment bounding the square and, similarly, see if you get a minimum L^2 close to zero. If so, the solution is an intersection point on that edge. This scheme works if the curve can only intersect an edge of a square at most once. There is no way, other than perhaps graphically, to determine multiple intersection points at an edge with any reliability.