Accrued illegal time

I am married to a USC. My children came to the US to visit with me, and while they were here my ex-husband decided it was an opportunity to take an extended overseas vacation, and I was unable to send them back to NZ within their visa time...their was noone to send them home to! I contacted the INS, advised them of the situation, and was told that I could include my children on my application. Two of the children decided to return home when their father finally arrived home so I did not complete their applications, they were here almost a year. Now, my children wish to visit with me again...I have read information regarding 'accrued illegal time', and all that I have read states that minor children are not subject to the ban on returning if they overstayed their visa. Does anyone know if this is absolutely correct? I am having nightmares about them arriving in LA and being refused entry. Any information would be greatly appreciated.

I am married to a USC. My children came to the US to visit with me, and while they were here my ex-husband decided it was an opportunity to take an extended overseas vacation, and I was unable to send them back to NZ within their visa time...their was noone to send them home to! I contacted the INS, advised them of the situation, and was told that I could include my children on my application. Two of the children decided to return home when their father finally arrived home so I did not complete their applications, they were here almost a year. Now, my children wish to visit with me again...I have read information regarding 'accrued illegal time', and all that I have read states that minor children are not subject to the ban on returning if they overstayed their visa. Does anyone know if this is absolutely correct? I am having nightmares about them arriving in LA and being refused entry. Any information would be greatly appreciated.