The ex is arguably no longer a tenant unless he pays some sort of compensation still.
The children are tenants. The older one (the adult child); give proper notice per your state's law to terminate the tenancy. If he fails to leave by the end of the time, file for eviction.
The younger one; contact cps and report the father has abandoned his child and you, not being the mother, want him gone.
If for some reason they won't remove the child, head to family court and file for child support. The father is liable for supporting his child. There is no reason you should shoulder it alone. I would suspect one of those two actions will get the father to take the child out of your house.
If the guy wants to continue to argue he has tenants rights, tell him his new monthly rent (which begins as soon as the law regarding prior notice allows) is somewhere high enough to pay your mortgage as well taxes utilities and enough for you to eat out at Ruth's Chris restaurant once a week.
Then in conjunction with the written notice of the rent increase give him a legally proper notice of termination of tenancy. If the rent due date comes before the termination date, if he doesn't pay, give a proper notice to pay or quit.
Then if all his stuff isn't out by the end of the termination date, file to evict.
Make sure all noticdes comply with the legal requirements both in form and time and means of delivery.