All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If you have confirmed that the chickens are slaughtered according to the Sharee'ah (i.e. slaughtered as Muslim do), then there is no harm in eating it even though the slaughterer is a Christian or Jew.
If you have confirmed that the way of slaughtering is unlawful, such as killing by electric shock or the like, then eating it is not permissible. Mentioning the Name of Allah at the time of slaughtering is not compulsory. The evidence for that as Imam Ash-Shawkaani said, is the verse {…and the food of those who were given the Scripture is lawful for you and your food is lawful for them…} [Quran 5:5]
'In the above verse Allah made it lawful for us to eat the meat slaughtered by the People of the Book in spite of the uncertainty whether they mentioned the Name of Allah or not.' [Nail Al-Awtar]
In short, if Christians and Jews slaughtered according to the Sharee'ah, it is permissible to eat such meat even if they do not mention the Name of Allah. If they do not slaughter the animals but they killed them by electric shock or any other way, then eating such meat is not permissible, even if the Name of Allah is mentioned. This is the opinion of the majority of Muslim scholars, and it is the correct one, Allah willing.
For more benefit on meat of animals slaughtered by the People of the Book, please refer to Fataawa 82323, 82065, 81234, 393961, 330952, and 82899.
Allah knows best.