It seems to me that the answer to your question is no, because of the following example (which came to my mind after reading Artie Prendergast-Smith's comment).
Consider a pencil $\lambda Q_1 + \mu Q_2$ of quartic surfaces in $\mathbb{P}^3$, and let $Z$ be its base locus, that in general will be a smooth curve of degree $16$. Blowing-up $Z$, we obtain a smooth rationally connected $3$-fold $X$ together with a map $\pi \colon X \to \mathbb{P}^1$, which gives to $X$ the structure of a fibration in $K3$ surfaces. If $F$ is the class of a fibre of $\pi$, the formula for the canonical class of a blow-up yields
$K_S=-F$. K_X=-F$.
So for every surface $S \subset X$ one has $(-K_S)^2 (-K_X)^2 \cdot S=0$.
This can be obviously generalized in any dimension, by considering a pencil of hypersurfaces of degree $n+1$ in $\mathbb{P}^n$ and blowing-up the corresponding base locus. In this way one obtain a smooth rationally connected $n$-fold $X$ with a fibration $\pi \colon X \to \mathbb{P}^1$ in Calabi-Yau varieties, and the anticanonical divisor of $X$ coincides with a fibre of $\pi$.
