Here's a proposal for a proof, though I can't finish it.

• A morphism in a category is a regular epimorphism if it coequalizes some pair of morphisms.
• An object $P$ in a category is projective if $\mathrm{Hom}(P,X)\to \mathrm{Hom}(P,Y)$ is surjective for every regular epimorphism $X\to Y$.
• An object $K$ in a category is compact if $\mathrm{Hom}(K,-)$ takes filtered colimits to filtered colimits of sets.
• An object $X$ in a category is irreducible if the only retracts of $X$ are itself and the initial object.

A self-equivalence of a category will preserve the these conditions.

Claim: $\mathbb{Z}[x]$ is (up to isomorphism) the unique irreducible compact projective in commutative rings.

What I do know (I think):

• Regular epis in $\mathrm{Rings}$ are exactly the surjective ring homomorphisms.
• Compact objects in $\mathrm{Rings}$ are exactly the finitely presented rings.

Since polynomial rings are certainly projective, this means that the compact projective objects in $\mathrm{Rings}$ are precisely the retracts of the $\mathbb{Z}[x_1,\dots,x_n]$s.

My intuition is that retracts of a polynomial ring are always isomorphic to a polynomial ring. In which case $\mathbb{Z}[x]$ would be the only irreducible compact projective. But I could be completely wrong about that.