Am I permitted to claim that I am a Christian to get a job?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.

It is not permissible for a Muslim to utter words of disbelief, except in necessity. Allaah The Almighty Says (what means): {Except for one who is forced [to renounce his religion] while his heart is secure in faith. But those who [willingly] open their breasts to disbelief, upon them is wrath from Allaah, and for them is a great punishment.} [Quran 16:106]

Necessity in this context refers to inevitable death, losing a body part and the like. Instead, it is better for a Muslim to die of hunger rather than to utter words of disbelief. A Muslim who willingly says a word that entails disbelief becomes a disbeliever thereby, even if he does not have inner belief of it or is joking or does so for the sake of money or property.

Therefore, you are not permitted to accept such a mean offer or to even think about it. What is the value of any gain if Allaah The Almighty is displeased with you? What is the value of any loss if Allaah The Almighty is pleased with you? When Ka‘b ibn Maalik and three of the Companions of the Prophet did not join the battle of Tabook, the Prophet ordered that nobody should communicate with them. Ka’b was outcast for fifty days until he admitted that he was desperate and suffered from people's harsh attitude towards him.  At that awkward moment, a man, similar to the person who made you the offer, came to him from Ash-Shaam to deliver him a message from the King of Ghassaan, who was Christian. In this message, the King invited Ka‘b to join him so that he would be granted wealth and authority.  Ka‘b burnt the message after reading it. [Al-Bukhari]

Honorable brother, do not hesitate to be like Ka‘b . Allaah The Almighty Says (what means): {And whoever fears Allaah - He will make for him a way out. And will provide for him from where he does not expect.} [Quran 65:2-3]

Indeed, apostating from Islam is the gravest sin, and renders the  person's good deeds worthless. Indeed a person could die immediately after having given up Islam and he will be an apostate  when he meets his Lord. Allaah The Almighty will be angry with him and will send him to Hell where he will abide forever. You must fear Allaah The Almighty, supplicate to Him and seek refuge in Him, for the keys to sustenance are in His Hand. 

Allaah Knows best.