The statement in question says that there exists SOME n for which its square is strictly less than n. If you show a counter example, there is nothing barring the fact that you simply chose the wrong n. A theorem would be something like "for all integers n, ." Notice the universal quantifier on this statement. To show it is false, you simply have to show that "for all" does not hold true. But as I emphasized above, the statement you provided was not a universal statement. It had an existential quantifier. Therefore, you need to know what it is you are countering. You would need to counter that actual n for which it is supposed to be true. However, you don't know which n that could be. It can be any positive integer. Therefore, you need to prove it is false for all possible choices of positive integers.

To think about it differently, suppose you run the dialog in your head.

"There exists a positive integer for which its square is strictly less than it." Yeah, but the square of 1 is not less than 1. "That is true, but there still exists a positive integer for which it is true."

You see the problem there? You would reasonably charge the person with "well, which positive integer is it?!" It is like saying "there exists intelligent life on other planets." Yeah, but look at Mars, we don't see any intelligent life. "Yeah, so? It's still out there somewhere." To prove that charge false, you would have to show that on every planet out there in the universe, there is no intelligent life. Of course, that is probably impossible. However, proving the statement you provided is not so impossible. In fact, it is quite easy to demonstrate that it cannot be true of the positive integers.