Eisenstein Criterion Shift Conditions

Good question! The answer is no, but it's surprisingly hard to produce either an explanation why or a counterexample without introducing some algebraic number theory.

The short story, which is still not so short, is that if $f(x)$ is, say, a monic integer polynomial of degree $n$, we can associate to it an integer $\Delta$ called the discriminant of the number field it generates, and if Eisenstein's criterion works for a prime $p$ on $f(x)$ or any translate of it (we can set $a = 1$ without loss of generality), then $p^{n-1}$ must divide $\Delta$, and it's possible to write down examples of cubic polynomials $f(x)$ (so that $n-1 = 2$) such that $\Delta$ is squarefree.

The long story involves a concept in algebraic number theory called ramification. See this PDF for some details, although you might need to crack open a textbook on algebraic number theory first.

In particular, from Wikipedia I learn that if $f(x) = x^3 - x^2 - 2x - 8$ then the discriminant is $-503$, which is the negative of a prime, so Eisenstein's criterion cannot be used on any translate of $f$.