n is an odd number

In the case of an odd size, start with a 5x5 defective square with the hole in the bottom left corner. Then you can build a border that's three units wide on the top and right sides and has no holes in it. Consider the case of extending a 5x5 defective square to an 8x8 defective square:

+-----+-----+-----+-----+
|     |     |     |     |
|     |     |  +--+--+  |
| 3x2 | 3x2 |  |     |  |
|     |     +--+--+  +--+
|     |     |     |  |  |
+-----+-----+--+  +--+  |
|              |  |     |
|              +--+-----+
|              |        |
|  previous    |   2x3  |
|    5x5       |        |
|   square     +--------+
|              |        |
+--+           |   2x3  |
   |           |        |
   +-----------+--------+

To extend a 7x7 square to a 10x10 square, use exactly the same pattern, but just add an extra 2x3 square onto each end of the new border. You can see that this will work to extend any defective square with n odd and the hole in the bottom left corner.