Yes, it does hold in positive characteristic. You can show that the degree of $c_n(X)$ equals its Euler characteristic with respect to étale cohomology with coefficients in $\mathbb{Q}_{\ell}$, where $\ell$ is a prime different from the characteristic ( http://mathoverflow.net/questions/80697/top-chern-class-in-positive-characteristic ), and this is has the required behavior under étale covers ( http://mathoverflow.net/questions/41576/behaviour-of-euler-characteristics-in-characteristic-p-for-finite-etale-covers ).
[Edit:] the equality can be proved directly; in fact the proof is much easier. Since $\pi\colon Y\to X$ is étale, the tangent bundle $T_Y$ is the pullback $\pi^*T_X$. By functoriality of Chern classes, $c_n(Y) = \pi^*c_n(X)$ (at the level of Chow groups). But it easy to see that the composite $\pi_*\pi^*$ from the Chow group of $X$ to itself is just multiplication by $d$.
Yes, it does hold in positive characteristic. You can show that the degree of $c_n(X)$ equals its Euler characteristic with respect to étale cohomology with coefficients in $\mathbb{Q}_{\ell}$, where $\ell$ is a prime different from the characteristic ( http://mathoverflow.net/questions/80697/top-chern-class-in-positive-characteristic ), and this is has the required behavior under étale covers ( http://mathoverflow.net/questions/41576/behaviour-of-euler-characteristics-in-characteristic-p-for-finite-etale-covers ).