He can try to do so. However he has to prove to the Judge that it is not in the best interests of the children to be around this person. It would be up to the discretion of the Judge. I do not think alone that being an undocumented immigrant would rise to the level of removing the children or having to kick your fiance out of the house. I would also suggest that your fiance seek advice from an immigration attorney. If he becomes documented and a legal resident then the problem will not exist.
Generally, a judge will not step in and evaluate the decision about whom a parent spends time with unless it can be shown that the parent's significant other is a danger to a child. There is nothing in your post which suggests that your ex husband is raising anything other than your fiancee's immigration status, which has nothing to do with the child. I agree with the previous poster that your ex husband raising this issue with a judge may present problems for your fiancee and that your fiancee should seek the advice of an immigration attorney to address this issue.