Generic local rings of reduced schemes

Choose an affine neighborhood $U=\mathrm{Spec}A$ of $\eta$, corresponding to $\mathfrak{p}$. If $\mathfrak{q}⊊\mathfrak{p}$ were prime, the closure in $X$ of the point of $U$ corresponding to $\mathfrak{q}$ would be irreducible and strictly contain the closure of $\eta$: its intersection with $U$ contains the point $\mathfrak{q}$, whereas the closure of $\eta$ does not. This contradicts maximality of the component. Thus $\mathfrak{p}$ is minimal.

The primes of $A_{\mathfrak{p}}$ correspond to primes of $A$ contained in $\mathfrak{p}$, so its only prime is its maximal ideal. The nilradical of a ring is the intersection of its prime ideals; consequently that maximal ideal is the nilradical. Reducedness of $X$ makes this local ring reduced, so the ideal is zero. A local ring with zero maximal ideal is a field.