Continuing the study on Schur analysis in the slice hyperholomorphic setting started in [the first author et al., Integral Equations Oper. Theory 72, No. 2, 253–289 (2012; Zbl 1258.47018)], this paper is a large study on realizations of slice hyperholomorphic generalized contractive and positive functions. At the beginning, some facts on the classical case are recalled. Elements of the classical case about Schur analysis, negative squares, slice hyperholomorphic functions are given, and in the second section some facts on quaternionic Pontryagin spaces are discussed, especially negative squares and kernels, realizations and slice hyperholomorphic functions. In the third section, operator-valued slice hyperholomorphic functions, differentiability, the \(\star\)-product of slice hyperholomorphic functions with values in a two sided quaternionic Banach algebra are analyzed and a useful property of a slice hyperholomorphic extension is given. Similarly as the Hardy space of holomorphic functions on a half-plane, the Hardy space of slice hyperholomorphic functions on the half-plane of quaternions with real positive parts is considered and its corresponding properties are studied in the fourth section of the paper. Also, the Blaschke factors and Blaschke products in the half-space of quaternions are introduced, and it is proved that the operator of multiplication by a Blaschke factor is an isometry. In the fifth section, a proof of the quaternionic version of the Schauder-Tychonoff fixed point theorem is given and this extension is used to prove an invariant subspace theorem for contractions in quaternionic Pontryagin spaces. The sixth section deals with the study of kernels with a finite number of negative squares and an associated reproducing kernel Pontryagin space. This reproducing kernel Pontryagin space is taken as the state space for a realization theorem obtained in the seventh section. Also, an example of a characteristic operator function of a quaternionic non anti-selfadjoint operator is given. In the last section of the paper, a realization for generalized positive functions is given. Also, using the existence of a square root of a positive operator in a quaternionic Pontryagin space, a positive function associated to a pair of anti-self-adjoint operators is defined, which plays an important role in models for pairs of anti-selfadjoint operators.