Do oriented null cobordant manifolds admit spin structures?

Let $M$ be an oriented null cobordant manifold.

Since $M$ is oriented its first Stiefel-Whitney class vanishes.

Since $M$ is null cobordant all of its Stiefel-Whitney numbers vanish.

Is it known if this implies that the second Stiefel-Whitney class vanishes so that $M$ admits a spin structure? Or are there known counterexamples?

No: Take your favourite orientable non-spin manifold $M$. Then $M\times S^1$ also does not admit a spin structure as the second stiefel whitney class does not vanish. But this manifold is nullbordant: It is the boundary of $M\times D^2$, where $D^2$ is the disc.