Odd Squares

[Taken from Schoenfeld p. 265 with additions by Stephanie Wilcox]

Show that the sum of consecutive odd numbers, starting with 1, is always a square. For example:

1+3+5+7=16 = 4^2

What would make this possible for all squares? Does it work for all squares? Find a deduction that shows that this problem does or does not work for all squares.