Answer to old chestnut: remove 2 matches to leave 2 squares:

 ___ ___
|       |
|       |
|    ___|
|   |   |
|   |   |
|___|___|

Aye, there's the rub! You have to leave one big square and one small one or you'll never solve this puzzle.

For the second puzzle:

 ___ ___ ___
|   |   |   |
|   |   |   |
|___|___|___|
|   |       |
|   |       |
|___|       |
|   |       |
|   |       |
|___|___ ___|

Note that the two solutions are inconsistent with one another... the former solution counts the big square which has a smaller square contained within it, while this one ignores it and thus counts 6 squares instead of 7.

Alternate answer to the second puzzle:

 ___ ___ ___
|           |
|           |
|    ___ ___|
|   |   |   |
|   |   |   |
|   |___|___|
|   |   |   |
|   |   |   |
|___|___|___|

This solution is consistent with the first, and counts the big square. IMHO, this makes it more correct than the solution given above.