This is basically Exercise 10.1.5(c) in Hodges's Model theory. First, a reminder of some definitions:
Let $\lambda$ be a cardinal, and let $\Sigma$ be a finitary first-order signature. A $\lambda$-saturated $\Sigma$-structure is a $\Sigma$-structure $A$ such that the following holds:
- If $X$ is a subset of $A$ and $\left| X \right| < \lambda$, then every complete $1$-type over $X$ with respect to $A$ is realised by an element of $A$.
Let $A$ be a $\Sigma$-structure. A definition for an element $a$ with parameters in a subset $X \subseteq A$ is a first-order formula $\phi (y, \vec{x})$ over $\Sigma$ and a finite sequence $\vec{b}$ of elements in $X$ such that $A \vDash \phi[y, \vec{b}] \leftrightarrow y = a$. A definable element of $A$ over $X$ is an element for which there exists a definition with parameters in $X$.
Question. Suppose $A$ is an $\aleph_0$-saturated $\Sigma$-structure and $a$ is not definable without parameters. Why should there be an automorphism of $A$ that moves $a$?
Obviously, if there is such an automorphism, then there would have to be an element $a'$ in $A$ such that $a \ne a'$ but $a$ and $a'$ have the same type with respect to $A$, since automorphisms preserve types. This is no problem because we can use a compactness argument to build an elementary extension of $A$ where there is such an $a'$, and then use $\aleph_0$-saturation to realise that $a'$ in $A$. Then it suffices to find an automorphism of $A$ that moves $a$ to $a'$... but it is not clear how I am supposed to do this.
Theorem 10.1.8(a) gives a solution in the case where $A$ itself is countable, because then we can just enumerate all the elements of $A$ and do a kind of back-and-forth argument. If $A$ is $\aleph_0$-big then we could apply Exercise 10.1.4(a). What about the general case?
Addendum. Here is the full text of exercise 10.1.5:
Let $\lambda$ be an infinite cardinal, $A$ a $\lambda$-saturated structure and $X$ a set of fewer than $\lambda$ elements of $A$. Show (a) if an element $a$ of $A$ is not algebraic over $X$, then infinitely many elements of $A$ realise $\textrm{tp}_A(a/X)$, (b) if an element $a$ of $A$ is not definable over $X$, then at least two elements of $A$ realise $\textrm{tp}_A(a/X)$, (c) $\textrm{ACL}(X) = \textrm{acl}(X)$ and $\textrm{DCL}(X) = \textrm{dcl}(X)$.
Parts (a) and (b) are quite easy exercises in using the compactness theorem.