Well, let me say what I know so far.

For monic quadratic polynomials it's necessary and sufficient that both roots be real and one be positive with absolute value at least the other.  This requires no complicated argument: the characteristic polynomial of $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is $x^2 - (a + d)x + (ad - bc)$.  Since $a, d \geq 0$ it's necessary that at one root has positive real part at least as large as the absolute value of the real part of the other, and since $b, c \geq 0$ it's necessary that $(a + d)^2 \geq 4ad \geq 4(ad - bc)$.  This is sufficient because we can set $c = 1$.

For general polynomials, I believe a theorem of Berstel implies that 1) the radius of convergence of $1/x^n P(1/x)$ must occur as a positive real pole $r$, and 2) any other pole $s$ with $|s| = r$ has the property that $s/r$ is a root of unity.  On the other hand polynomials such as the polynomial with roots $5, 5, 3 + 4i, 3 - 4i$ don't have this property even though they satisfy the non-negativity condition.