Fix an integer $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $ω$. We consider the specialized skew hook Schur polynomial $\text{hs}_{λ/μ}(X,ωX,\dots,ω^{t-1}X/Y,ωY,\dots,ω^{t-1}Y)$, where $ω^k X=(ω^k x_1, \dots, ω^k x_n)$, $ω^k Y=(ω^k y_1, \dots, ω^k y_m)$ for $0 \leq k \leq t-1$. We characterize the skew shapes $λ/μ$ for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials. Then we give a combinatorial interpretation of $\text{hs}_{λ/μ}(1,ω^d,\dots,ω^{d(tn-1)}/1,ω^d,\dots,ω^{d(tm-1)})$, for all divisors $d$ of $t$, in terms of ribbon supertableaux. Lastly, we use the combinatorial interpretation to prove the cyclic sieving phenomenon on the set of semistandard supertableaux of shape $λ/μ$ for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee--Oh (arXiv: 2112.12394, 2021) for the cyclic sieving phenomenon on the set of skew SSYT conjectured by Alexandersson--Pfannerer--Rubey--Uhlin (Forum Math. Sigma, 2021).