Well, let me say what I know so far.
For monic quadratic polynomials it's necessary and sufficient that both roots be real. This requires no complicated argument: the characteristic polynomial of [a b] [c d] is x^2 - (a + d)x + (ad - bc), and since a, d ≥ 0 it's necessary that (a + d)^2 ≥ 4ad ≥ 4(ad - bc). On the other hand, 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.