Zero and one have the same quantifier-free type in the multiplicative integers

In the constant-free language with multiplication as its sole nonlogical symbol, every term in one free variable is $x^k$ for some $k\ge 1$. Every atomic formula $x^r=x^s$ holds at both zero and one. Hence every quantifier-free formula, being a finite Boolean combination of those atoms, has the same truth value at both points. They have the same quantifier-free type, and neither singleton is quantifier-free definable. Quantifiers distinguish them: $\forall y(xy=x)$ defines zero, while $\forall y(xy=y)$ defines one. These formulas do not rely on named zero or one constants.