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.

It has been proved in the preprint (page 6) by Renee Hoekzema 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. It is quite similar to the one Mike Miller given here without the induction procedure.

The author suggests there is a more direct proof generalizing the one from Milnor-Stasheff without using the Euler class. I am not sure it might be. The paper actually proved much more and I found it really interesting.