About a variation of the primitive root idea.

Question

Let $n $ be called a half primitive root mod ($m$) if $n^{\phi(m)/2} = 1$ mod($m$) and for any $t $ with $1 < t < \phi(m)/2$, $m$ does not divide $(n^t - 1)$. So the order of $n$ mod($m$) is $\phi(m)/2$. Also $n, n^2 , n^3 , .., n^{(p-1)/2}$ are all distinct and they are all the quadratic residues mod($m$) in some order. All integers that are the products of at least two distinct prime divisors have a half primitive root , is this true?