Balrogs

[Sharkû]

Could they have taken the form of a child of Eru?

That appears unlikely. At any rate, it seems downright impossible after the final incarnation of the Balrogs.

"If the Balrogs at the Battle of the Powers were perhaps only ‘clothed’ rather than fully incarnate, they would have been able to reassume hröar when Melkor returned later. However, their later deaths were more permanent due to incarnation, or an increased degree of such. They were inherently weaker Maiar than Sauron – who himself may only have been able to reincarnate so many times because of his Ring – and could not re-embody themselves as he did." (obloquy)