Are there any symplectic but not complex Calabi Yau manifolds in real dimensions 4 and 6?

First of all, the notion Symplectic CalabiYau is quite new. A few persons who use it (including myself) usually mean by this symplectic manifolds with $c_1=0$, (this is just to make sure that we speak about the same thing) In real dimension $4$ we know for the moment only two types of symplectic CalabiYau manifolds  $K3$ surfaces and $T^2$ bundles over $T^2$. These manifolds have as well the structure of a complex manifold with a nonvanishing holomorphic volume form. It is conjectured that there are no other symplectic Calabi Yau manifolds in dimension $4$. In real dimension six there are quite a lot of symplectic CY manifolds coming from the twistor construction (you can check here: http://arxiv.org/abs/0802.3648), and some of them do have a complex structure, but this is not known for all of them. At the same time, probably you know that in dimension $2n\ge 6$ the following question is open: Question. Is it true that every manifold $M^{2n}$ that has an almost complex structure $J$ has as well a holomorphic structure homothopic to J? This is an old question and apparently no one has an idea of how to answer it. Now, the answer to your question in dimension $6$ depends on what you mean by a complex CalabiYau. This notation is not used in math literature. If by such a manifold you mean a complex manfiold with $c_1=0$, then you would not be able (for the moment) to get any example in dimension $6$ where the answer to your question is no (because the above Question is open). On the other hand, if by complex CalabiYau you mean a complex manifold with a nonvanishing holomorphic volume form, then the answer to your question is yes, an example is given in http://arxiv.org/abs/0905.3237. There is a symplectic Calabi Yau 6manifold in this paper, that has $b_3=0$, hence it can not have a holomorphic volume form of top degree for any complex structure. One can construct further such examples. 

