Let $R$ commutative ring with unity, $S\subseteq R$ subring, $p$ minimal prime ideal of $S$. Show there exists a minimal prime ideal $q$ in $R$ with the property that the contraction $q^c=q\cap S=p$.
First of all, I am not sure whether the minimality of $q$ refers to the prime ideals in $R$ or to the prime ideals with such contraction property, if the latter is the case, then we only need to show the existence of such prime ideal, minimal one would be given by Zorn's lemma. Second, if we drop the minimality condition, then the proposition clearly doesn't hold ($\mathbb{Z}\subseteq\mathbb{Q}$ for instance), so this condition must be crucial here.
It is enough to find a single prime ideal $q$ satisfying the condition; since any smaller prime ideal will also satisfy the condition (by minimality of $p$), we can find the desired minimal prime using Zorn.
Let $q$ be an ideal of $R$, maximal with respect to the property that $q\cap S \subset p$ (such a $q$ exists because $(0)$ satisfies the condition). It is not so hard to show that such a $q$ is prime, so we are done by minimality of $p$.
(I don't want to write out the last proof in full, since it's equivalent to an easier one with localizations: we have $R_p\neq 0$, so it has a prime ideal; let $q$ be its preimage in $R$, which has the desired property because everything in $S\setminus p$ gets mapped to a unit)