No.
If you take for $X$ the subvariety $X\subset \mathbb A^2(\mathbb C)$ defined by $xy=0$, the first projection $\phi: X\to \mathbb A^1_\mathbb C=Spec(\mathbb C[t])$ is not flat because the dimension of the fibers $\phi^{-1}(t)$ jumps at $t=0$, nevertheless $\phi^{-1}(0)\cong \mathbb A^1(\mathbb C)$ is a complete intersection.
Edit
It is easy to see purely algebraically that $\phi$ is not flat.
Indeed, the dual map $\phi^*:\mathbb C[t]\to \mathbb C[x,y]/(xy):t\mapsto \bar x$ is not flat because over a PID (like $\mathbb C[t]$) the flat modules are exactly the torsion-free modules and here the element $\bar y\in \mathbb C[x,y]$ is a torsion element: $t\cdot \bar y=\bar x \bar y=0$