The best characterization I can currently work out is: Fix a proper (not necessarily nontrival) coset $H$. Form a disjoint union of sets of the following form: Choose a subgroup $H'$. (not necessarily proper) that strictly contains $H$. Choose a coset of $H'$, and for each coset of $H$ within that coset choose a representative. Then this disjoint union is not good.

Proof: Let $k$ be the smallest element of $H$, then the elementary cyclotomic polynomial of $k$ divides the polynomial of each such set of coset representatives, so it divides the polynomial of a disjoint union.

I do not know if this characterization is complete, but it seemed too long to fit in the comments.