I think the equivalence

"every unital ring homomorphism $A\to B$ with $B\neq 0$ maps $a$ to a non-unit $\Longleftrightarrow$ $a$ is nilpotent"

cannot be true (though $\Longleftarrow$ holds, of course). Otherwise, it would yield that whenever $a$ is nilpotent, then so is $uav$ for any $u$ and $v$ from $A$, but this is not satisfied in the ring $\mathbb{Z}\left\langle X,Y\right\rangle / \left(X^2\right)$ (the ideal is two-sided). The counterexample is $u=Y$, $a=X$, $v=1$ (while $a=X$ is nilpotent, $uav=YX$ isn't).