Is it permissible to eat chicken slaughtered by a non-Muslim (Buddhist or irreligious person) in a chicken shop, if I pronounce the Basmalah and the Takbir while he is slaughtering it?

The one who slaughters must be a Muslim or a person of the Book (Christian or Jew). As for the slaughter of those other than them—such as an idol worshipper, an atheist, or a Buddhist—it is not Islamically permissible, and their slaughtered animal is not to be eaten, even if a Muslim pronounces the name of Allah over it.

Al-Marghinani mentioned that the slaughter of a person of the Book is disliked (makruh) but permissible if a Muslim orders it, unlike the Magian (Zoroastrian), whose slaughtered animal is not permissible.