I certainly don't know of any restrictions placed on muslims going into Churches. The Church of the Holy Sepulcher was invaded by Amaleks, er I mean Palestians, for example.
That said, I CAN tell you from the Orthodox Jewish perspective, which may be the same, that Orthodox Jews are not supposed to go into Churches because that could be seen as an endorsement, even accidental, of the Christian religion, which is viewed --- by some Jews --- as Polytheistic, and thus a violation of the most important mitva, not to worship any God, but God.
I am presuming the problem --- if there is legitamately is a problem --- doubtful with the ACLU --- the muslim objection is the same.
> I certainly don't know of any restrictions placed on
> muslims going into Churches.
I looked it up, as well as I could in a few minutes. Apparently, the restriction is only on (as the story noted) churches with images, paintings, sculpture and such like.
Harkens back to the idolatry provisions which, like so much else, are the same for everybody but everybody interprets them differently.