Based on the deformed nucleon distributions obtained from the constrained Skyrme-Hartree-Fock-Bogolyubov calculation using different nuclear symmetry energies, we have investigated the effects of the neutron skin and the collision geometry on the yield of free spectator nucleons as well as the yield ratio $N_{\rm{n}}^{}/N_{\rm{p}}^{}$ of free spectator neutrons to protons in collisions of deformed nuclei at RHIC energies. We found that tip-tip (body-body) collisions with prolate (oblate) nuclei lead to fewest free spectator nucleons, compared to other collision configurations. While the $N_{\rm{n}}^{}/N_{\rm{p}}^{}$ ratio is sensitive to the average neutron-skin thickness of colliding nuclei, and is thus a good probe of the symmetry energy, it is affected by the polar angular distribution of the neutron skin in different collision configurations. We also found that the collision geometry effect can be as large as 50% the symmetry energy effect in some collision systems. Due to the particular deformed neutron skin in ²³⁸U and ⁹⁶Zr, the symmetry energy effect on the $N_{\rm{n}}^{}/N_{\rm{p}}^{}$ ratio is enhanced in tip-tip ²³⁸U+²³⁸U collisions and body-body ⁹⁶Zr+⁹⁶Zr collisions compared to other collision configurations in the same collision system. Our study may shed light on probing deformed neutron skin by selecting desired configurations in high-energy collisions with deformed nuclei.