Let $X=\mathbb{C}^3$, $Q=V(x^2-yz)$, $H_1=V(y)$, $H_2=V(z)$. The hyperplanes $H_i$ are tangent to $Q$ along a line. Let $U$ be the complement of $Q\cup H_1\cup H_2$. Is the fundamental group $\pi_1(U,\mathrm{pt})$ isomorphic to $\mathbb{Z}^3?$
It helps that all of the varieties involved are conical, so we may consider the real picture and see that this is the same as the fundamental group of a sphere $S^5$ without sections by a pair of 4-planes and the cone, but I don't see what the intersection with the cone looks like in relation to the two copies of $S^3$ we throw out.
This fundamental group should also be isomorphic to the fundamental group of a section by a general 2 plane (more specifically, transversal to each stratum of a Whitney stratification of $X\setminus U$). This reduces the problem to computing the fundamental group of the plane $\mathbb{C}^2$ without two intersecting lines and a smooth quadric tangent to each of these lines, but I was unable to compute this either.