To minimize the expression $x^2 + y^2 + z^2 + mxy + nxz + pyz$ with the constraint $xyz = 8$, we will follow a systematic approach rooted in mathematical optimization techniques.

### Case $(a)$: $m = n = p = 2$

1. **Substitute for $z$ using the constraint**:
Since $xyz = 8$, express $z$ in terms of $x$ and $y$:
$$z = \frac{8}{xy}.$$

2. Substitute into the expression:
Insert $z = \frac{8}{xy}$ into $x^2 + y^2 + z^2 + mxy + nxz + pyz$:
$$x^2 + y^2 + \left( \frac{8}{xy} \right)^2 + 2xy + 2x \left( \frac{8}{xy} \right) + 2y \left( \frac{8}{xy} \right).$$

3. Simplify the expression:
$$x^2 + y^2 + \frac{64}{x^2y^2} + 2xy + \frac{16}{y} + \frac{16}{x}.$$

4. Minimize the expression:
Use symmetry (since $m = n = p = 2$) and consider $x = y = z$. Given $xyz = 8$, we have:
$$x = y = z = \sqrt[3]{8} = 2.$$

5. Calculate the minimum value:
$$3x^2 + 3 \cdot 2 \cdot x^2 = 9x^2 \quad \text{with} \quad x = 2,$$
$$= 9 \cdot 2^2 = 36.$$

### Case $(b)$: Arbitrary $m, n, p$

1. Apply Lagrange multipliers:
To find the critical points of $f(x, y, z) = x^2 + y^2 + z^2 + mxy + nxz + pyz$ subject to the constraint $g(x, y, z) = xyz - 8 = 0$, set:
$$\nabla f = \lambda \nabla g.$$

The gradients are:
$$\nabla f = (2x + my + nz, 2y + mx + pz, 2z + nx + py),$$
$$\nabla g = (yz, xz, xy).$$

2. Solve the equations:
Solving the system:
$$2x + my + nz = \lambda yz, \quad 2y + mx + pz = \lambda xz, \quad 2z + nx + py = \lambda xy.$$

3. Assuming symmetry (or cyclic permutation):
$$x = y = z = \sqrt[3]{8} = 2.$$

4. Verify minimization point by calculation:
Rearrange to achieve symmetry or substitution help find reasonable point usage. The minimum often occurs for:
$$z = \sqrt[3]{4m}.$$

### Conclusion

For both the cases, we find the minimum value given when symmetry holds or crafting is optimized about relationships considering their modifier influences. Ultimately, for arbitrary $m, n, p$, the equality condition is achieved when:

$$\boxed{z = \sqrt[3]{4m}}$$

This identifies the point optimally considering conditions specified and constraints bound within problem requirements.