= Solution
This class is axiomatisable in the original language. For every positive integer $n$, let $\sigma_n$ say
$$
\text{``there are at least $n$ maximal elements''}
\quad\lor\quad
\text{``there are at least $n$ minimal elements''.}
$$
Both clauses are <first-order sentence>[first-order]. If one of the two sets is infinite, every $\sigma_n$ holds. Conversely, if both sets were finite, choosing $n$ larger than both cardinalities would make $\sigma_n$ false. Hence the <first-order theory> consisting of the poset axioms and all the $\sigma_n$ has exactly the required models.
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