It's obvious how to divide a square into 4 smaller squares. That doesn't directly help you with this claim (which starts at n=6), but perhaps the division into four might be useful in an indirect way.
Try dividing one of the squares into smaller squares. How does this change the number of squares?
For n=6, we can cut up the square as follows, where the side length of the big square is twice that of the smaller squares.
6 Squares ________ |__|__|__| | |__| |_____|__|
Can you generalize the construction for n=6 to work for any other number of subsquares?
Make a table of solutions you've found for different values of n.
It's convenient if all the solutions look very similar. However, it's also ok if you need to use different methods for different values of n, e.g. one method for the odd numbers and one for the even numbers.