Legally, am I required to send my children to an absentee parent who has reappeared in their lives?

A general principles you should be aware of: If he is in contempt (for non-payment of $17,000 in support...or any amount), it does not give you the right to also be in contempt by not allowing visitation. My typical response to these types of situations is to say that 'yes' you are legally required to send the children. If you refuse to do so, however, it is unlikely that law enforcement would become involved and force the children to go with them; although I have actually seen that happen. More likely than not, if he tries to get law enforcement to escort him to receive the children and you refuse to let them go, the officers will tell him it is a civil matter. His remedy will then be to file a motion for contempt asking that you be held in contempt of court for violating the court order. In my mind, there is always a better way to handle this. According to your recitation of the facts, the papers read that he can have the children 2 days a week, "upon mutual agreement". That seems strange to me. In any event, it would appear that you do not mutually agree to this. I would sit down with him and try to discuss a re-introductory period with the children. If you truly feel that the children may be harmed or in danger by letting them go with their father, hire an attorney and modify the terms of the final order.

It would be unusual if all your divorce order states regarding visitation is "up to 2 days a week, upon mutual agreement ," but if that is all it says, are you in "mutual agreement" for the visitation? It sounds like you are not. You should probably take your divorce order to an attorney for a review to ensure you have the correct understanding.

You are required to follow the order of the court. You can not decide to change the order on your own. The only way a court order may be changed is by a judge.

Child support and visitation are not connected. You can't deny visitation just because the other person isn't paying their child support. However, if he hasn't had contact in a while, you can use that as a factor in getting your visitation agreement modified. It is obvious you want to do what is best for your kids, and fighting with your ex over custody isn't a good way to go about that. Work with your ex on coming to an agreement where you can work together towards introducing him back into their lives. Something like dinner this week, an afternoon together next week, a full day the week after that, then two days. It may help to convince him by saying you'll agree to let him work up to paying back the child support he owes in a similarly way, a little extra this week, a little more the week after. As an attorney above said, he is in contempt on the child support, but don't go into contempt on visitation. If it goes to court, you want to be seen as the one trying to work things out for the kids. Actually trying to work things out is the best way to look like the one trying to work things out.