15. Given that $a, b, c$ are distinct integers, the minimum value of $4\left(a^{2}+b^{2}+c^{2}\right)-(a+b+c)^{2}$ is
$$
\begin{array}{l}
4\left(a^{2}+b^{2}+c^{2}\right)-(a+b+c)^{2}=3\left(a^{2}+b^{2}+c^{2}\right)-2(a b+b c+a c) \\
=a^{2}+b^{2}+c^{2}+(a-b)^{2}+(b-c)^{2}+(a-c)^{2} \geqslant 1+0+1+2 \cdot 1+2^{2}=8
\end{array}
$$

The equality holds when $(a, b, c)=(-1,0,1)$. Therefore, the minimum value of the original expression is 8.