Is it permissible to work as a butcher in a non-Muslim country where they do not slaughter the meat in the manner prescribed in Islam?

Praise be to Allah.

Firstly: 

What is required is to denounce evil and combat it, and not approve of it, and to prevent people from getting involved in it or falling into it, because of the general meaning of the verse in which Allah, may He be exalted, says (interpretation of the meaning):

“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allah. Verily, Allah is Severe in punishment”

[al-Maa’idah 5:2].

And because the Prophet (blessings and peace of Allah be upon him) said: “Whoever among you sees an evil action, then let him change it with his hand [by taking action]; if he cannot, then with his tongue [by speaking out]; and if he cannot, then with his heart – and that is the weakest of faith.” Narrated by Muslim (49). 

An-Nawawi (may Allah have mercy on him) said: 

The words of the Prophet (blessings and peace of Allah be upon him), “… let him change it”, form a command which indicates that it is obligatory according to the consensus of the ummah; the proof for the obligation to enjoin what is good and forbid what is evil is based on the Qur’an, Sunnah and scholarly consensus. End quote. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: If a man helps others in disobeying Allah, he is sinning, because he is helping in sin and transgression. Hence the Prophet (blessings and peace of Allah be upon him) cursed alcohol, the one who presses (the grapes, etc), the one for whom it is pressed, the one who carries it, the one to whom it is carried, the one who sells it, the one who buys it, the one who pours it, the one who drinks it and the one who consumes its price. Most of these people, such as the one who presses it, the one who carries it, and the one who pours it, are helping in drinking it. Hence it is forbidden to sell weapons to one who will use them in a forbidden fight, such as fighting the Muslims or fighting in the event of fitnah (turmoil, civil war). (22/141-142) 

Once this is understood, then slaughter in a manner other than that which is prescribed in Islam is haraam, so it is not permissible for the Muslim to do this himself, and it is not permissible for him to help others in it, whether that is by doing the slaughtering or disposing of the meat after it has been slaughtered by selling or buying and so on. 

Al-Bukhaari (2236) and Muslim (1581) narrated from Jaabir ibn ‘Abdullah (may Allah be pleased with him) that he heard the Messenger of Allah (blessings and peace of Allah be upon him) , during the year of the Conquest when he was in Makkah: “Allah and His Messenger have forbidden the sale of alcohol, dead meat, pork and idols.”. 

Al-Haafiz ibn Hajar (may Allah have mercy on him) said: 

Ibn al-Mundhir and others narrated that there was consensus on the prohibition on selling dead meat (that which died without being slaughtered in the manner prescribed in Islam), but they made an exception from that in the case of fish and locusts.

End quote from Fath al-Baari (4/424). 

Anything that has not been slaughtered in the manner prescribed in Islam comes under the heading of “dead meat”, such as that which died of natural causes and was not slaughtered properly. 

Ibn al-Qayyim (may Allah have mercy on him) said: 

With regard to the prohibition on selling dead meat, that includes everything that may be called dead meat, whether it died of natural causes or was slaughtered in a manner that does not make it permissible..

End quote from Zaad al-Ma‘aad (5/749). 

See also the answer to questions no. 7489 and 85191

And Allah knows best.