Well that's called blackmail as you cannot threaten to do something negative to a person in exchange for them doing something you want them to do; you need to call the police and report her for blackmail. And since you know she is in the US illegally you might want to call ICE, too.

Now, as for child support, was there ever a court order for child support? Being illegal in the US does not stop a person from going to court to get a child support or custody order for their children. If she has waited this long to go to court over it, a judge will most likely only go back a year or two, not a full 10 years, in order to determine back child support; and she would be entitled to that. Any money you have paid her so far would be considered gifts unless you were ordered to pay child support and/or were paying through the proper channels (ie, not depositing it straight into her account).

We have been here for 10 years, she has only been here for 2. There hasn't been any court ordered child support filed in the US. Won't the support be effective as of the date filed?

If the children were born in another country, and she and the children were living in another country all that time, it's the other country that would have jurisdiction over child support for that period of time, not the US.

Now that she's here, she can certainly file for child support going forward. It's a civil matter and they don't much care about her or the children's legal status.

Her threats are petty, but putting that aside, Is there a reason your husband won't petition for his children? To do so, they may need to return to their country. Seems a small thing to do for them, really.