Consider the ideal $$I := (xy - 4, z^2 - zy - y) \subseteq \mathbb{Z}[x,y,z].$$ I'd would like to know if it is prime.
If it is prime, then I believe I can conclude that $\mathbb{Z}[x,y,z]/I$ is a flat module over $\mathbb{Z}$. If it is not prime, then can we show that $\mathbb{Z}[x,y,z]/I$ is flat (or not flat) over $\mathbb{Z}$ using a different method?
Indeed $I$ is prime. We have a ring injection $$\begin{align} \mathbb{Z}[x,y,z]/I & = \mathbb{Z}[x,4/x,z]/(z^2-(4/x)z-(4/x)) \\ & \hookrightarrow\mathbb{Z}[x,1/x,z]/(xz^2-4z-4). \end{align}$$
The latter morphism is the same as $$\mathbb{Z}[x,4/x]\oplus \mathbb{Z}[x,4/x] \to \mathbb{Z}[x,1/x]\oplus \mathbb{Z}[x,1/x]$$ and in fact injective.
$xz^2-4z-4$ is irreducible in $\mathbb{Z}[x,z]$. For, if not, it factorizes as $(xz+P(x))(z+Q(x))$ where $P(x)+xQ(x)=-4, P(x)Q(x)=-4.$ From the second equality we obtain $P(x),Q(x)\in \mathbb{Z}\setminus \{0\}$, contradicting the first one. Thus $\mathbb{Z}[x,y,z]/I$ is an integral domain.
Since $\mathbb{Z}\cap (xz^2-4z-4)=0$ in $\mathbb{Z}[x,z]$, $\mathbb{Z}\to \mathbb{Z}[x,y,z]/I$ is an injection and in particular $ \mathbb{Z}[x,y,z]/I$ is a torsion-free module over $\mathbb{Z}$. This is a characterization of flatness for modules over a PID.