If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?

Let me propose the following counterexample:

Take $X = \overline{\mathbb Q}$ the one-point compactification of $\mathbb Q$. This space is not Hausdorff, since $\mathbb Q$ is not locally compact (the problem is $\infty$).

Now let $f: \overline{\mathbb Q} \to \overline{\mathbb Q}$ be a continuous function, and let $x\in \overline{\mathrm{Fix}(f)}$ be an arbitrary point in the closure of $\mathrm{Fix}(f)$.

Case I: Suppose $\infty \in \mathrm{Fix}(f)$. Then either $x = \infty$ or we must have that the restriction $f|_{\mathbb Q}: \mathbb Q \to \overline{\mathbb Q}$ is continuous. But then also $x \in \overline{\mathrm{Fix}(f|_\mathbb{Q})} \subset \mathrm{Fix}(f|_\mathbb{Q}) \cup \{\infty\}= \mathrm{Fix}(f)$.

Case II: Now suppose $\infty \notin \mathrm{Fix}(f)$ and $x\ne \infty$. Then there is a convergent sequence $x_n \to x$ with $x_n\in \mathrm{Fix}(f)$. But then by continuity: $f(x) = \lim_{n\to \infty} f(x_n) = \lim_{n\to \infty} x_n = x$.

So there is only one case left:

Can we have $x=\infty \in \overline{\mathrm{Fix}(f)}$ but at the same time $\infty \notin \mathrm{Fix}(f)$?

If this were the case, then $\mathrm{Fix}(f)$ would definitely not be compact. But this implies that there must be a sequence in $x_n \in \mathrm{Fix}(f) \subset \mathbb Q$ without a convergent subsequence - that is: no convergent subsequence in $\mathbb Q$!

But then such a sequence has a subsequence converging to $\infty$, which implies that $\infty \in \mathrm{Fix}(f)$.

Hoping I haven't made some silly mistake, this concludes the argument that this $X$ is indeed a counterexample.

While I think that Sam's counterexample is fine, I would like to elaborate on ccc's comment to Sam's answer.

Recall that a topological space $X$ is called Fréchet if for every $A\subseteq X$ and every point $x\in \bar{A}$ there exists a sequence of points of $A$ converging to $x$.

Observe that if $X$ is Fréchet with unique sequential limits, then $\text{Fix}(f)$ is closed for every continuous $f:X\to X$. Indeed, if $X$ is such a space, $f:X\to X$ is continuous and $x\in\overline{\text{Fix}(f)}$, then by Fréchetness of $X$ we find a sequence $(x_n)_{n=1}^\infty$ of points of $\text{Fix}(f)$ converging to $x$. Then $f(x)=f(\lim_{n\to\infty} x_n)=\lim_{n\to\infty}f(x_n)=\lim_{n\to\infty}x_n=x$ by continuity of $f$ and uniqueness of sequential limits in $X$. This means that $x\in\text{Fix}(f)$ and hence $\text{Fix}(f)$ is closed.

To give a positive answer to the question we want to find a non-Hausdorff Fréchet space with unique sequential limits. The one-point compactification of the rationals suggested by Sam is such a space. Another example is Example 6.2 in S.P. Franklin, Spaces in which sequences suffice II, Fund. Math. 61 (1967), which is available here.

This is a remark related to the question.

Theorem : let $X$ be a topological space. Suppose that for every space $Y$ and pair of maps $f,g : Y \rightarrow X$, the subset $\{ y \in Y : f(y)=g(y) \}$ is closed. Then $X$ is Hausdorf.

It is easy to prove by considering the $2$ projections $X \times X \rightarrow X$. In algebraic geometry, a (pre)variety which satisfies this axiom is called separated (see for example Milne's note on page 61 http://www.jmilne.org/math/CourseNotes/ag.html). This notion was introduced because algebraic varieties are not Hausdorff, and if we work with separated varieties, then maps are determined on dense subset.