Edit
----

For `r=Sqrt[53]`, there are `148` squares in the curve, and there are `56` outside the curve, and `16*16-148-56=52` on the curve.

```
Clear["Global`*"];
lines0 = 
 Flatten[{Table[
    Line[{{i, j}, {i, j} + {1, 0}}], {i, -8, 7}, {j, -8, 8}], 
   Table[Line[{{i, j}, {i, j} + {0, 1}}], {i, -8, 8}, {j, -8, 7}]}];
p = {0, 0};
r = Sqrt[53];
range1 = Disk[p, r];
range2 = Annulus[p, {r, 4 r}];
intersec[range_] := Module[{lines, pts, edges, g, faces, squares},
   lines = 
    Cases[RegionIntersection[range, #] & /@ lines0, _Line, Infinity];
   pts = DeleteDuplicates[Flatten[lines[[;; , 1]], {2, 1}]];
   edges = 
    Table[UndirectedEdge @@ Flatten[FirstPosition[pts, #] & /@ d], {d,
       lines[[;; , 1]]}];
   g = Graph[edges, 
     VertexCoordinates -> Thread[Range@Length@pts -> pts]];
   faces = PlanarFaceList[g];
   squares = Select[faces, Length[#] == 4 &];
   {MeshRegion[pts, Polygon /@ squares], Length[squares]}];
Labeled[Graphics[{lines0, {FaceForm[], EdgeForm[Cyan], 
    range1}, {FaceForm[Yellow], EdgeForm[Red], 
    intersec[range2] // First}, {FaceForm[Green], EdgeForm[Red], 
    intersec[range1] // First}}], {Style[intersec[range1] // Last, 
   Green, 20], Style[intersec[range2] // Last, Darker@Yellow], 
  Style[2*8*2*8 - intersec[range1][[2]] - intersec[range2][[2]], 
   20]}]
```

[![enter image description here][1]][1]

**Original**
* Similar with https://mathematica.stackexchange.com/a/280656/72111, we list all of the segments, and intersection such lines with a region.
* Then we construct a `Planar Graph` and calculate all the 4-cycles.


```
lines0 = 
  Flatten[{Table[
     Line[{{i, j}, {i, j} + {1, 0}}], {i, -8, 7}, {j, -8, 8}], 
    Table[Line[{{i, j}, {i, j} + {0, 1}}], {i, -8, 8}, {j, -8, 7}]}];
Manipulate[Module[{range, lines, pts, edges, g, faces, squares},
  range = Disk[p, Sqrt[59]];
  (* range=Annulus[p,{3,Sqrt[59]}]; *)
  lines = 
   Cases[RegionIntersection[range, #] & /@ lines0, _Line, Infinity];
  pts = DeleteDuplicates[Flatten[lines[[;; , 1]], {2, 1}]];
  edges = 
   Table[UndirectedEdge @@ Flatten[FirstPosition[pts, #] & /@ d], {d, 
     lines[[;; , 1]]}];
  g = Graph[edges, 
    VertexCoordinates -> Thread[Range@Length@pts -> pts]];
  faces = PlanarFaceList[g];
  squares = Select[faces, Length[#] == 4 &];
  Labeled[
   Show[Graphics[lines0], 
    Graphics[{EdgeForm[Cyan], FaceForm[], range, EdgeForm[Red], 
      FaceForm[Green], MeshRegion[pts, Polygon /@ squares]}], 
    PlotRange -> 10], Framed@Length@squares]], {{p, {0, 0}}, Locator, 
  Appearance -> None}]
```
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/oVz9p.png
  [2]: https://i.sstatic.net/s8VRM.gif