By the Weierstrass Factorization Theorem, every entire function f can be represented as a product involving its zeroes.
Moreover, if $\{ a_n \}$ is the sequence of zeroes of $f$ then $\displaystyle f(z) = z\prod_{n=1}^{\infty} E_n(z/a_n)$ where $E_n$ is the elementary factor (a.k.a canonical factor) defined as $E_n(z) = (1-z)exp({\sum_{k=1}^{n} \frac{z^k}{k}})$
Putting in the conditions for positive squares, I got the function $\displaystyle f(z) = z\prod_{n=1}^{\infty} (1-\frac{z}{n^2})exp({\sum_{k=1}^{n} \frac{z^k}{k}})$. However, this diverges for large n because of the exponential term. What is the purpose of the exponential term and the canonical factors in this example?
If my solution or thinking is wrong, how would one go about creating an entire function that is zero at the positive squares?