The proofs I have seen for the hairy ball theorem all use either degree of a map defined in e.g. by homology or direct computations using stereographic projections in order to use homotopy arguments in $\mathbb R^2$.
Isn't there a trick to deduce the theorem from the Borsuk-Ulam theorem for the sphere $S^2$?
This is unlikely, since Borsuk-Ulam works for all $n$. Such an argument could not work for, say, the 3-sphere, since the latter can indeed be combed. In fact $S^3$ is parallelizable. This follows for example from the fact that $S^3$ is a Lie group, for example that of unit quaternions.
This is indirect, but maybe it will be helpful to you:
as pointed out in the paper "A Borsuk-Ulam Equivalent that Directly Implies Sperner’s Lemma" (also available here) by Nyman and Su, the Borsuk-Ulam theorem implies Sperner's lemma.
then "The hairy ball theorem via Sperner's lemma (also here) by Jarvis and Tanton use Sperner to prove the hairy ball theorem (for $S^2$, not for general $S^{2n}$). Perhaps their proof is due to Shashkin (see their references), but I haven't checked.
