The relation ${\sim}\subseteq A^2$ is called euclidic if for all $x,y\in A$ it follows that: $$x\sim y ~~\wedge ~~ x\sim z \implies y\sim z$$ The relation ${\sim}\subseteq A^2$ is called asymmetric if for all $x,y \in A$ it follows that: $$x\sim y \implies y\not \sim x$$
The task asks to check if: $\sim $ euclidic $\implies \sim$ not asymmetric.
Attempt: Suppose $\sim$ is both euclidic and asymmetric, for some $x,y\in A$ such that $x\sim y$ it follows that $x\sim y$ which by euclidic property implies that $y\sim y$, but then from asymmetric property it follows that $y\not \sim y$. So we arrive at $y\sim y$ and $y\not \sim y$, which is a contradiction.
However, if we take that $\sim$ is empty relation, then it is both euclidic and asymmetric, since both premises are trivially achieved. Now I wonder if empty relations count as relations.
The empty relation is indeed a relation, Your argument works assuming there exist $x,y\in A$ such that $x\sim y$. But, it does not apply to the empty relation, since in that case there do not exist any such $x$ and $y$. So indeed, the implication you are investigating is not always true: the empty relation is a counterexample.