Denote $E = a b^{2} + b c^{2} + c a^{2} - a^{2} b - b^{2} c - c^{2} a$. We have

$$\begin{aligned} E = & \left(a b c - c^{2} a\right) + \left(c a^{2} - a^{2} b\right) + \left(b c^{2} - b^{2} c\right) + \left(a b^{2} - a b c\right) = \\ & (b - c)\left(a c - a^{2} - b c + a b\right) = (b - c)\left(a (c - a) - b (c - a)\right) = (b - c)(c - a)(a - b) \end{aligned}$$

So, $|E| = |a - b| \cdot |b - c| \cdot |c - a|$. By hypothesis each factor from $|E|$ is a positive integer. We shall prove that at least one factor from $|E|$ is greater than 1. Suppose that $|a - b| = |b - c| = |c - a| = 1$. It follows that the numbers $a - b, b - c, c - a$ are odd. So, the number $0 = (a - b) + (b - c) + (c - a)$ is also odd, a contradiction. Hence, $|E| \geq 1 \cdot 1 \cdot 2 = 2$.