The original problem solves in the positive: there is a model of ZFC in which there exists a countable OD (well, even lightface $\Pi^1_2$, which is the best possible) set of reals $X$ containing no OD elements. The model (by the way, as conjectured by Ali Enayat at http://cs.nyu.edu/pipermail/fom/2010-July/014944.html) is a $\mathbf P^{<\omega}$-generic extension of $L$, where $\mathbf P$ is Jensen's minimal $\Pi^1_2$ real singleton forcing and $\mathbf P^{<\omega}$ is the finite-support product of $\omega$ copies of $\mathbf P$.
A few details. Jensen's forcing is defined in $L$ so that $\mathbf P =\bigcup_{\xi<\omega_1} \mathbf P_\xi$, where each $\mathbf P_\xi$ is a ctble set of perfect trees in $2^{<\omega}$, generic over the outcome $\mathbf P_{<\xi}=\bigcup_{\eta<\xi}\mathbf P_\eta$ of all earlier steps in such a way that any $\mathbf P_{<\xi}$-name $c$ for a real ($c$ belongs to a minimal countable transitive model of a fragment of ZFC, containing $\mathbf P_{<\xi}$), which $\mathbf P_{<\xi}$ forces to be different from the generic real itself, is pushed by $\mathbf P_{\xi}$ (the next layer) not to belong to any $[T]$ where $T$ is a tree in $\mathbf P_{\xi}$. The effect is that the generic real itself is the only $\mathbf P$-generic real in the extension, while examination of the complexity shows that it is a $\Pi^1_2$ singleton.
Now let $\mathbf P^{<\omega}$ be the finite-support product of $\omega$ copies of $\mathbf P$. It adds a ctble sequence of $\mathbf P$-generic reals $x_n$. A version of the argument above shows that still the reals $x_n$ are the only $\mathbf P$-generic reals in the extension and the set $\{x_n:n<\omega\}$ is $\Pi^1_2$. Finally the routine technique of finite-support-product extensions ensures that $x_n$ are not OD in the extension.
Addendum. For detailed proofs of the above claims, see this manuscript.
Jindra Zapletal informed me that he got a model where a $\mathsf E_0$-equivalence class $X=[x]_{E_0}$ of a certain Silver generic real is OD and contains no OD elements, but in that model $X$ does not seem to be analytically definable, let alone $\Pi^1_2$. The model involves a combination of several forcing notions and some modern ideas in descriptive set theory recently presented in Canonical Ramsey Theory on Polish Spaces. Thus whether a $\mathsf E_0$-class of a non-OD real can be $\Pi^1_2$ is probably still open.
Kanovei's addendum. With thanks to Ali for the 1401.3901 link above, I continue on Aug 21 with a simpler (?) model for a definable $\mathsf E_0$-class in $2^\omega$ with no definable elements. One may ask a similar Q wrt any reasonable Borel equivalence relation $\mathsf E$, of course, but with little chance to answer it immediately.
Fix a recursive enumeration $\{s_k:k\in\omega\}$ of all dyadic strings in $2^{<\omega}$, such that $s_0=\Lambda$ (the empty string).
Let $T$ be the tree of all dyadic strings of the form $\langle{k,0^m}\rangle$ and $\langle{k,0^m,1}\rangle$, where $k,m\in\omega$, plus the empty string $\Lambda$. Consider a ramified iterated ctble-support forcing extension $M$ of the constructible universe $L$ by a system of reals $x_t$, $t\in T$, such that $x_\Lambda$ is Silver over $L$ (Sacks will not work!!) and each $x_t$ is Sacks over $\{x_s:s\subset t\}$. Then in $M$, the reals $x_{\langle{k}\rangle}$ are the only minimal $L$-degrees over $x_\Lambda$ while the reals $x_{\langle{k,0^m}\rangle}$ and $x_{\langle{k,0^m,1}\rangle}$ (and the pairs $\langle{x_{\langle{k,0^{m'}}\rangle},x_{\langle{k,0^m,1}\rangle}}\rangle$, $m'>m$) are the only $L$-degrees over $x_{\langle{k}\rangle}$ incomparable to all $x_{\langle{\ell\rangle}}$, $\ell\ne k$.
Let, in $M$, $y_k=x_\Lambda+s_k$, where $+$ is the dyadic addition, that is $y_k(n)=x_\Lambda(n)$ whenever $s_k(n)=0$ and $y_k(n)=1-x_\Lambda(n)$ whenever $s_k(n)=1$. Then $y_0=x_\Lambda$ and $\{y_k:k\in\omega\}$ is the whole $\mathsf E_0$-class of $x_\Lambda$.
Now consider the submodel $N$ of $M$ generated by the reals
$x_\Lambda$
all reals $x_{\langle{k}\rangle}$, $k\in\omega$
all reals $x_{\langle{k,0^m}\rangle}$, $k\in\omega$
all reals $x_{\langle{k,0^m,1}\rangle}$, $k\in\omega$ and $y_k(m)=1$
Then in $N$, each subtree $T_k$ = all $L$-degrees of reals
$x_{\langle{k}\rangle}$,
$x_{\langle{k,0^m}\rangle}$,
$x_{\langle{k,0^m,1}\rangle}$, $y_k(m)=1$
codes $y_k$ by means of degrees of constructibility, while the whole forest codes the set $\{y_k:k\in\omega\}$ = the $\mathsf E_0$-degree of $x_\Lambda$, making the latter set analytically definable (lightface $\Sigma^1_7$, say) in $N$.
It takes some work to check finally that $x_\Lambda$ itself is not definable in $N$. This is based on the fact that the Silver forcing responsible for $x_\Lambda$ is invariant wrt the group $P=\prod_mP_m$, where $P_m$ is the $0-1$ flip at $m$th digit, in such a way that each Silver condition is invariant under $\prod_{m>n}P_m$ for a suitable $n$.
To conclude, this gives a model with an analytically definable $\mathsf E_0$-class (most likely not $\Pi^1_2$-definable), with no OD elements.
Further Kanovei's addendum of Aug 23. It looks like a clone of Jensen's forcing on the base of Silver's (or $\mathsf E_0$-large Sacks) forcing instead of the simple Sacks one leads to a lightface $\Pi^1_2$ generic $\mathsf E_0$-class with no OD elements. The advantage of Silver's forcing here is that it seems to produce a Jensen-type forcing closed under the 0-1 flip at any digit, so that the corresponding extension contains a $\mathsf E_0$-class of generic reals instead of a generic singleton. I am working on details, hopefully it pans out.
Further Kanovei's addendum of Aug 25. Yes it works, so there is a generic extension $L[x]$ of $L$ by a real in which the $\mathsf E_0$-class $[x]_{\mathsf E_0}$ is a lightface $\Pi^1_2$ (countable) set with no OD elements. I'll send it to Axriv in a few days.
Further Kanovei's addendum of Aug 29. arXiv:1408.6642