Can the custodial parent claim both children and the earned income credit on their tax return if the non-custodial parent is behind on child support but usually you each claim one child?

The custodial Parent has the right to claim both children, but in your case if the NC parent is behind they will automatically take it from their tax refund anyway. But as with any agreement if it states the NC gets to claim one then you can only claim one. If there is not an agreement in that way, then the custodial has the right. The arrears owed to the C has no affect on the claiming rights of another.