It is not true that global choice implies $V=OD$.
Global choice is the assertion that there is a class well-ordering of the universe. This is equivalent to the assertion that there is a global choice function, which selects from every non-empty set an element. It is not part of the axiom, however, that this class is definable. Rather, the class well-ordering of $V$ is simply one of the classes that is available in the second order part of the model, and there is no reason to suppose that all such classes are necessarily first-order definable, even from set parameters.
The axiom $V=HOD$, on the other hand, is equivalent to the assertion that there is a definable global choice function class, or equivalently, a definable well-ordering of the universe.
So the critical difference is in the question of whether the global well-ordering is definable or not. It might be interesting to note that there can be a global well-ordering that is definable from set parameters, and so still counts as a class in ZFC, without having $V=HOD$, simply because those parameter may be essential, and it may not be possible to define the order without them. This is what happens, for example, after adding a generic Cohen real over $L$, since in $L[c]$ we do not have $V=HOD$, and in fact $\text{HOD}^{L[c]}=L$ by the homogeneity of the forcing, but nevertheless one can use $c$ as a parameter to define in $L[c]$ a well-ordering of the universe, essentially the usual $L$-order relativized to $c$.
The proof that NGBC is conservative over ZFC is not difficult. Given any model $M$ of ZFC, one first forces global choice by class forcing: consider the class partial order consisting of well-ordered sequences of longer and longer set length. This forcing is $\kappa$-closed for every $\kappa$, and hence adds no new sets, but the generic filter $G$ provides a class well-ordering of $M$. Now, to form the GBC model $M[G]$, one considers as classes all the classes that are definable from set parameters and this new global well-order class. One can verify that the resulting model satisfies all the GBC axioms (and this amounts essentially to the verification that this kind of class forcing works as expected). Since the sets of $M[G]$ are the same as $M[G]$, it follows that any statement that is true in every GBC model is also true in every model of ZFC, and so the extension is conservative.
Best Answer
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.
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