Solution:

Let $P(x) = x^3 + x + 1,$ so the product is $\prod_{k=0}^{6} P(\omega^k),$ where $\omega^0, \ldots, \omega^6$ are all seventh roots of unity. Since $x^7 - 1 = \prod_k (x - \omega^k),$ writing $P(x) = (x - \beta_1)(x - \beta_2)(x - \beta_3)$ and swapping the order of the double product gives $$\prod_{k=0}^{6} P(\omega^k) = \prod_{j=1}^{3} \prod_{k=0}^{6} (\omega^k - \beta_j) = \prod_{j=1}^{3} \bigl(-(\beta_j^7 - 1)\bigr) = \prod_{j=1}^{3} (1 - \beta_j^7).$$

For a root $\beta$ of $P,$ repeatedly using $\beta^3 = -\beta - 1$ gives $\beta^4 = -\beta^2 - \beta,$ $\beta^5 = -\beta^2 + \beta + 1,$ $\beta^6 = \beta^2 + 2\beta + 1,$ and $\beta^7 = 2\beta^2 - 1.$ Hence $1 - \beta^7 = 2(1 - \beta)(1 + \beta),$ and $$\prod_{j} (1 - \beta_j^7) = 2^3 \prod_j (1 - \beta_j) \prod_j (1 + \beta_j) = 8 \cdot P(1) \cdot \bigl(-P(-1)\bigr) = 8 \cdot 3 \cdot 1 = 24.$$

So the requested product equals $24.$