Rigid pentagons and rational solutions of $s^4+s^3+s^2+s+1=y^2$

Not true I'm afraid. There are, in fact, an infinite number of rational solutions.

The curve is a quartic with a rational point $(0,1)$, and is thus birationally equivalent to an elliptic curve, which has genus $1$. Faltings' Theorem only applies if the genus is strictly greater than $1$.

The equivalent elliptic curve is $v^2=u^3-5u^2+5u$ with $s=(2v-u)/(4u-5)$. The point $(0,0)$ is the only finite torsion point and we can take $(1,1)$ as a generator.

The rational solutions you give come from small multiples of the generator. Larger examples are $-20965/43993$ and $-761577/1404304$, but you can get larger and larger solutions.