Whatever else they are Freemasons are NOT a Christian group, in fact as brotherhoods go they are united in spite of Religion are certainly not because of it. To become a Mason you have to believe in a God which is not the same thing.
Only in the minds of conspiracy theorists are there any connections between the Freemasons and the Illuminati, makes for great fiction and writers like myself use it in our stories but there is no proof that the Illuminati still exist.
Even at their peak they were as it says on the tin, 'The Illuminati' The Enlightened Ones and the only reason they went underground and became a secret society was to prevent prosecution from the church who branded all who disagreed with the churches or biblical explanation for things as heretics and subject to the Inquisition, torture and possible death.
PS I am not a Freemason, I do not believe in a God, any of them, and because of that I could be considered that I am one of the Illuminati.