Putnam 2016 B2

Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2 = 2025$ and $2025-2016=9$ is a perfect square. (Of the positive integers between 1 and 10, only 6 and 7 are not squarish.)

For a positive integer $N$, let $S(N)$ be the number of squarish integers between 1 and $N$, inclusive. Find positive constants $\alpha$ and $\beta$ such that $$\lim_{N \to \infty} \frac{S(N)}{N^\alpha} = \beta,$$ or show that no such constants exist.