It has been proved in the [preprint][1] (page 6) that the vanishing of the $w_{n}(M)$ implies $\chi(M)$ is even. The proof uses the fact that a symplectic vector space over $\mathbb{F}_{2}$ has even dimension. 

The author suggests there is a more direct proof generalizing the one from Milnor-Stasheff without using the Euler class. My guess is that it would be similar to the proof Mike Miller given above. 

  [1]: https://arxiv.org/pdf/1704.06607.pdf