Does the system function H(z) for a filter always have symmetry above and below the real x-axis?
No it does not (always posses symmetry).

However, for real systems whose coefficients $a_k ~,~ b_k$ of system function, 

$$H(z) = \sum_{n} h[n] z^{-n} = \frac{\sum_{k=0}^{M}b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} $$ are real, then accordingly its roots (which are the poles and zeros of $H(z)$) are either real or complex-conjugates. And because of this conjugation, the poles (or zeros) on the Z-plane posses the vertical symmetry as you've recognised.

On the contrary, for complex systems whose system function $H(z)$ is a complex polynomial of complex coefficients, we cannot constraint the roots of denominator and numerator of $H(z)$ to be real or complex-conjugate, and hence they can posses non-symmetric poles and zeros...