rectangles = Rectangle /@ Tuples[Range[-8, 7, 1], {2}]
Count[rectangles, _?(Not @ RegionDisjoint[Circle[{0, 0}, Sqrt[59]], #] &)]
60
Count[rectangles, _?(RegionWithin[Disk[{0, 0}, Sqrt[59]], #] &)]
156
Update: To exclude rectangles "just touching" the circle, we can use
Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
RegionIntersection[Circle[{0, 0}, Sqrt[59]], #] &)]
60
Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
RegionIntersection[Circle[{0, 0}, Sqrt[53]], #] &)]
52