Fundamental issues associated with the application of the nuclear-electronic orbital (NEO) approach to hydrogen transfer systems are addressed. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated with molecular orbital methods. The positions of the nuclear basis function centers are optimized variationally. In the application of the NEO approach to hydrogen transfer systems, the hydrogen nuclei and all electrons are treated quantum mechanically. Within the NEO framework, the transferring hydrogen atom can be represented by two basis function centers to allow delocalization of the proton vibrational wavefunction. In this paper, the NEO approach is applied to the [He-H-He]+ and [He-H-He]++ model systems. Analyses of technical issues pertaining to flexibility of the basis set to describe both single and double well proton potential energy surfaces, linear dependency of the hydrogen basis functions, multiple minima in the basis function center optimization, convergence of the number of hydrogen basis function centers, and basis set superposition error are presented. The accuracy of the NEO approach is tested by comparison to grid calculations for these model systems.