Universal Taylor series on doubly connected domains with respect to every center

We prove that there exist universal Taylor series -in the sense of Nestoridis- in the complement of a square with respect to every center. Furthermore, for a weaker notion of universal Taylor series due to Luh and Chui and Parnes, we prove that there exist universal Taylor series in the complement of the closed unit disk with respect to every center. Overconvergence phenomena with respect to different centers have been first investigated by W. Luh (Analysis 6 (1986) 191-207).