The partial order of statements $\leq$ the axiom of choice is simply the lower-cone below AC in the [Lindenbaum algebra](https://en.wikipedia.org/wiki/Lindenbaum%E2%80%93Tarski_algebra) of ZF statements. Thus, in particular, it is a Boolean algebra, where in this case, the equivalence class of AC is $1$. It follows that *every* nontrivial statement $\varphi$ that is strictly below AC forms a maximal antichain with its relative complement $\text{AC}\wedge\neg\varphi$. So of course, every nontrivial statement is part of a finite maximal antichain. (Asaf had noted a few instances of this in his answer.)

Meanwhile, since it is known, assuming the consistency of ZF, that there are infinitely many inequivalent statements weaker than AC (such as choice for families of various-sized finite sets, or DC for increasingly long transfinite sequences), it follows that the Boolean algebra is infinite. From this, it follows that there must be infinite antichains, simply because every infinite Boolean algebra has an infinite antichain.