Added: The later comment to the question requires the integers to be positive. So the following isn't an example.

There is a trivial construction for even $N$: From the $2N$ integers $\{\pm1, \pm3,\ldots, \pm(2N-1)\}$ pick a subset $A$ of size $N$ such exactly one of $k$ or $-k$ is in $A$. (So there are $2^N$ such choices of $A$). Then the product of the integers in $A$ equals the product of its complement.

Probably, up to scaling, this is the only construction. (Verified for all $N\le12$.)

A necessary condition for an example is that there are positive integers $a,b$ such that $\prod_{i=0}^{2N-1}(a+ib)$ is a square. Differently phrased (divide by $b^{2N}$), we ask for rational solutions of the hyperelliptic curve $Y^2=\prod_{i=0}^{2N-1}(X+i)$. According to a conjecture by Sander (see Nontrivial rational points on Erdős-Selfridge curves), there are no solutions except a trivial one which does not meet your positivity requirement.