Given a Borel equivalence relation $E \subset [X]^2$ with countable equivalence classes (let's say generated by the action of $\mathbb{F}_2$ on $2^\omega$ via some convenient coding.) Is it necessarily the case that any selection of representatives from each equivalence class, witnesses the existence of a non-measurable subset of $X$?
(if my assumption that it's generated by an $\mathbb{F}_2$-action is too strong, feel-free to ignore it.)
No.
Think about the relation $x-y\in\mathbb{Z}$ on $\mathbb{R}$; this has very simple transversals, but also has countable equivalence classes.
An $\mathbb{F}_2$-on-$2^\omega$ example, unless I misunderstand, is the equivalence relation "are equal on all but possibly the first bit." Again, this has a very simple transversal.