Answer: 8

We make use of two key facts:
First, $n^2 -1 = (n-1)(n+1)$.
Second, when a square number is factorised, each prime factor appears an even number of times.
 
Now $2^2 -1 = 1\times 3$.
We next get a prime factor $3$ with $4^2-1= 3\times 5$.
We next get a prime factor $5$ with $6^2-1= 5\times 7$.
We next get a prime factor $7$ with $8^2-1= 7\times 9$.
 
As $9=3^2$, it does not require any further factors. Hence we need $n\geq 8$. Checking the product $n=8$, we get
$(2^2 -1)(3^2 -1)(4^2 -1)(5^2 -1)(6^2 -1)(7^2 -1)(8^2 -1)$
$$\eqalign{
&=1\times 3\times2 \times4 \times3 \times5 \times4\times6 \times5\times7 \times6\times8 \times7\times9 \cr
&=2\times8 \times 3 \times 3\times4\times4\times5\times5\times6\times6\times7\times7\times3\times3 \cr
&=4\times4\times3\times3\times4\times4\times5\times5\times6\times6\times7\times7\times3\times3 \cr
&= (4\times3\times4\times5\times6\times7\times3)^2}$$
So in fact $n=8$ is sufficient, and is thus the minimum.