Im looking for non-UFD rings such that factoring of any element of that ring into irreducibles leads to either all factorizations squarefree or all factorizations squareful.
Thus let $n$ be an element of that ring then either $a^2 b c ... = A^2 B C ...= $ etc where the variables are distinct irreducibles. Or we get $x y z ... = X Y Z ...= etc$ with the variables distinct irreducibles. But we cannot get $a^2 b c ... = s d f ...$
This means if we have factorization of $n$ where at least one irreducible element divides $n$ twice than all factorizations are squareful. And if we have a factorization of $n$ where no irreducible element divides $n$ twice than any other factorization of $n$ lacks an irreducible than devides $n$ twice.