jmc

The absolute value of the product of two numbers is the product of their absolute values, so we can write $$|x^2-16|=|(x+4)(x-4)|=|x+4||x-4|.$$Since $|x^2-16|$ is written as the product of two positive integers, it is composite unless one of the integers is $1$. Solving $|x+4|=1$, we observe that either $x+4=1$ or $x+4=-1$, which give solutions of $x=-3$ and $x=-5$. Similarly, solving $|x-4|=1$ we find $x=3$ or $x=5$. Among the possible solutions $\{-5,-3,3,5\}$, only $\{-3,3\}$ yield a prime value for $|x+4||x-4|$. Therefore, the product of the integer values of $x$ for which $|x^2-16|$ is a prime number is $\boxed{-9}$.