$1$. Claim: $\varnothing$ is an antichain.
$Proof$: Suppose $\varnothing$ is not an antichain. Then $\exists$ a pair $x, y \in \varnothing$ such that $x$ and $y$ are comparable. Contradiction: $\varnothing$ is empty. Thus $\varnothing$ is an antichain.
$2$. Claim: $\varnothing$ is a chain.
$Proof$: Suppose $\varnothing$ is not a chain. Then $\exists$ a pair $x, y \in \varnothing$ such that $x$ and $y$ are incomparable. Contradiction: $\varnothing$ is empty. Thus $\varnothing$ is a chain.
What am I doing wrong here? Or is $\varnothing$ both a chain and antichain?
The logical error is the conclusion of the two proofs. Your specific definitons of a chain and an antichain together do not cover all the possibilities, so "is not a chain" does not necessarily imply "is an antichain." Using the specific definitions you've given would imply that $\emptyset$ is neither a chain nor an antichain.
If you changed the definition of, for example, antichain to be "not a chain" then the two would cover all possibilities. One of your proofs would no longer work and the other would give the correct answer.