ZZ Ceti stars are pulsating white dwarfs with a carbon-oxygen core build up during the core helium burning and thermally pulsing Asymptotic Giant Branch phases. Through the interpretation of their pulsation periods by means of asteroseismology, details about their origin and evolution can be inferred. The whole pulsation spectrum exhibited by ZZ Ceti stars strongly depends on the inner chemical structure. At present, there are several processes affecting the chemical profiles that are still not accurately determined. We present a study of the impact of the current uncertainties of the white dwarf formation and evolution on the expected pulsation properties of ZZ Ceti stars. Our analysis is based on a set of carbon-oxygen core white dwarf models with masses $0.548$ and $0.837 M_{\sun}$ derived from full evolutionary computations from the ZAMS to the ZZ Ceti domain. We have considered models in which we varied the number of thermal pulses, the amount of overshooting, and the $^{12}$C$(α,γ)^{16}$O reaction rate within their uncertainties. We explore the impact of these major uncertainties in prior evolution on the chemical structure and the expected pulsation spectrum. We find that these uncertainties yield significant changes in the $g$-mode pulsation periods. We conclude that the uncertainties in the white dwarf progenitor evolution should be be taken into account in detailed asterseismological analysis of these pulsating stars. ; Comment: 7 pages, 8 figures. To be published in Astronomy and Astrophysics