It has been proved in the preprint (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.