Poincaré-Hopf Theorem for manifolds with boundary in Milnor's book

Question

I've just read Milnor's topology from the differentiable viewpoint, though I haven't been able to figure out the case of manifolds with boundary. Milnor says that the problem is that in this case the tubular neighborhood $ N_\epsilon $ will no longer be smooth but only $ C^1 $, so that the extension on the neighborhood of the vector field defined on the manifold will be only continuous near the boundary of the manifold. Milnor says also that it is enough to show that the differentiability assumptions are not really necessary. I wonder how to do so.