An elementary way to show that the determinant is non zero 
Show that the determinant of the matrix
\begin{pmatrix} a && -c && -b \\
  b && a - 2c && -c -2b \\ c && b && a -2c
 \end{pmatrix} is non zero for all integers $a,b,c$ where $abc \ne 0$

There is an interesting way to do this by using integral domains. It is easy to see that the polynomial $t^3 + 2t + 1$ is prime in $\mathbb{Z}[t]$ so the ring $\mathbb{Z}[t]/\langle t^3+2t+1\rangle$ is an integral domain and isomorphic to the ring $\{a + bu + cu^2 \vert \text{$a,b,c$ are integers}\}$ where $u$ is a root of the equation $t^3 +2t + 1 =0$.
Now consider integers $a,b,c$ and $x,y,z$ such that $abc \ne  0$ and $(a +bu + cu^2)(x + yu + zu^2) = 0$. These are element of an integral domain so we must have that $x = y = z = 0$ since at least one of $a,b$ and $c$ are nonzero. But expanding the above equation we get a system of linear equations in terms of $x,y$ and $z$.
$$
\begin{aligned} 
ax- cy -bz &= 0 \\ 
bx+ (a-2c)y + (-c-2b)z &= 0\\ 
c x+b y+ (a-2c)z &= 0 
\end{aligned} 
$$
If the determinant is $0$ then this equation will have a nontrivial solution which is not possible.
This strategy requires the use of integral domains which is an abstract tool.
Are there any elementary ways to solve this? It gets messy when we try to do row operations or try to expand the determinant.
Update : After  Carl Schildkraut's and JimmyK4542's answer.
A curious observation:
We can also prove that the element $t^3 + rt + 1$ is prime in $\mathbb{Z}[t]$ for $r \ne 0, -2$. Therefore by similar arguments
we can say that the determinant of the matrix
\begin{pmatrix} a && -c && -b \\
  b && a - rc && -c -rb \\ c && b && a -rc
 \end{pmatrix} is non zero for all integers $a,b,c,r$ where $abc \ne 0$ and $r \ne 0,-2$.

This might be equivalent to what you did, but I'll go ahead and write this as an answer anyway.
Let $X = \begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & -2 \\ 0 & 1 & 0\end{bmatrix}$. Then, $X^2 = \begin{bmatrix}0 & -1 & 0 \\ 0 & -2 & -1 \\ 1 & 0 & -2\end{bmatrix}$, and so, $$Y := \begin{bmatrix}a & -c & -b \\  b & a - 2c & -c -2b \\ c & b & a -2c\end{bmatrix} = aI+bX+cX^2.$$
The eigenvalues of $X$ are the three roots $\lambda_1,\lambda_2,\lambda_3$ of $\lambda^3+2\lambda+1 = 0$.
Since $Y = aI+bX+cX^2$, the eigenvalues of $Y$ are $a+b\lambda_k+c\lambda_k^2$ for $k = 1,2,3$.
Now, suppose $\det(Y) = 0$ for some integers $a,b,c$ with $abc \neq 0$. Then, $0$ must be an eigenvalue of $Y$, and so, $a+b\lambda_k+c\lambda_k^2 = 0$ for some $k \in \{1,2,3\}$. But by using the quadratic formula, we have $\lambda_k = \dfrac{-b \pm \sqrt{b^2-4ca}}{2c}$.
So now you just need to prove that none of the three roots of $\lambda^3+2\lambda+1 = 0$ can be expressed in the form $\dfrac{-b \pm \sqrt{b^2-4ca}}{2c}$ for non-zero integers $a,b,c$. This just requires showing that $\lambda^3+2\lambda+1$ is irreducible in $\mathbb{Z}[\lambda]$, which only requires the rational root theorem and checking that $\pm 1$ are not roots.
