Please inform us about the following things:Did the Authors of al-Kutub al-Sittah ( Bukhari , Muslim , etc.) follow the school of thought of any Imam (Shafi'e, Maliki, Hambali, Hanafi, etc.)? If yes, what school did they follow?Please provide the reference look that quotes the Imams saying that we should follow the correct Hadith whenever we find it, and that we should not follow their opinion if it is against Sunnah.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Authors of al-Kutub al-Sittah ( Bukhari , Muslim , Abu Dawood , al-Tirmizi , al-Nasa'i and Ibn Majah ) - May Allah's Mercy be upon them - were not followers of any of the four schools of thought ( Abu Hanifa , Malik , al-Shafi'e and Ahmad - May Allah's Mercy be upon them-). They were Mujtahid Imams (qualified scholars, great jurists). They had their special viewpoints and opinions that were based on their understanding from the Qur'an and Sunnah.
But, according to writings of the authors of al-Kutub al-Sitta, there is no approved evidence that testifies their being followers of any of the four schools of Fiqh even though some Shafi'e Imams consider al-Bukhari and al-Tirmizi from among the followers of al-Shafi'e , as well as some Hanbali Imams consider al-Nasa'i and Abu Dawood as followers of their school of Fiqh.
As for the stand of the Mujtahid Imams in general and those of the four schools of thought, concerning any contradiction between a Hadith and any of their opinions, we advise you to consult "Iqaz al-Himam" book by al-Fallani and the preface of the book of "Sifat Salat al-Nabi" by al-AlBani (May Allah's Mercy be upon him); it is translated into English.
However, here is a well-known saying that is ascribed to Abu Hanifa and al-Shafi'e : 'It is part and parcel of my school of thought to follow every sound Hadith'.
al-Shafi'e said: 'If anyone finds that there is some contradiction between an opinion of mine and a sound Hadith, one has to disregard my opinion'.
Malik said: 'Everyone's sayings are subject to be followed or rejected except those of Allah's Messenger'.
Ahmad said: 'Do not follow my opinions without recognizing evidence nor those of Malik or al-Shafi'e , but consult the Qur'an and Sunnah directly'.
In simple words, the above words are addressed to jurists and qualified scholars who can make Ijtihad (independent opinions) depending on the Qur'an and Sunnah, but other people are advised to follow an Imam without any form of bigotry.
Allah knows best.