The kids, definitely JJ is a resident of Idaho because Lori ended up a resident of Idaho.
The instant Lori enrolled JJ in a public school she established JJ (and herself) as a resident of Idaho. Under Idaho Statutes:
An individual who has a place of abode in Idaho and is present in Idaho for other than a temporary or transitory purpose is deemed to reside in Idaho.
An individual is present in Idaho only for a temporary or transitory purpose if he does not engage in any activity or conduct in Idaho other than that of a vacationer, seasonal visitor, tourist, or guest.
Idaho can claim jurisdiction over JJ (and almost certainly Tylee) at a minimum because his last known residency is Idaho. If Lori wants to argue otherwise then she must prove it and the only way to prove it is to reveal where JJ is because he is certainly not with Lori in Hawaii. Arguments of jurisdiction are not predicated on the State having the burden of proof. Each side has to try to prove their case. Idaho has the establishment of a residency by renting the townhouse and enrolling JJ in public school. Lori has to prove Idaho lacks jurisdiction by proving where JJ (and likely Tylee) is.
The State of Idaho can, as authorized by statute, initiate a case on the basis of:
Reason to believe that a child has been abused, neglected or abandoned, or subject to conditions or circumstances which would reasonably result in abuse, abandonment, or neglect
Lori being in Hawaii without her children is a circumstance that can reasonably be construed to have resulted in neglect and/or abandonment. Thus, the intervention by the State of Idaho and the order by the court to produce the children. The purpose of the order to produce the children almost certainly states that it is to ascertain whether or not the children have been abused or neglected. Notice that the requirement is not that the State has to prove the children were abused, neglected, or abandoned only that there is reason to believe that is the case. Lori is free to refute such allegations and a judge will decide who prevails. It is not a criminal case so the burden of proof does not rest solely on the State.
She doesn't get to avoid this due to "parental rights". States have rights too. Idaho has an interest in the welfare of children and has constitutionally valid laws for intervening on behalf of children in adversarial actions against parents. Judges get to decide the best interests of children when two parties - in this case the State of Idaho and Lori - cannot agree on what is in the best interest of the involved children. That libertarians believe that their own interests carry more weight than the state's interest means little in the scheme of things and much more often than not serves to make their situation worse than it would otherwise be.
Keep in mind that the order to Lori to produce the children is not from an action commenced by the Woodcocks. They only filed a Guardianship case days after Lori was served. There was already a case by the State of Idaho in progress that we have not seen any of the docket to even know when it began since Child Welfare cases are not public information in Idaho and we only were given what the Court decided the State Attorney could release. That is why Lori's battle is with the State of Idaho (i.e. the People of Idaho). Idaho statutes only give legal standing to State Attorney's to file Child Welfare cases so that is how I am certain this is a case by the State of Idaho against Lori.
The Idaho Supreme Court has deemed that it is in the best interests of children that child welfare cases are handled speedily and Idaho statutes and court rules have been promulgated to the achieve this. Failure to meet deadlines or compliance with orders may result in a default judgment and thus dragging out the process is not possible. Pro Se litigants, for example, are warned that delay tactics they often employ in civil cases will not achieve their aims and can lead to a default judgment. The end result of a default is the establishment that the State's allegations are true and the defendant agrees those allegations are true. That is, defendants cannot appeal in order to re-litigate the facts once they have defaulted. It is not known if a default judgment has been entered since we can't see the docket but that is the risk Lori takes with her non-compliance. I suspect she has an attorney litigating but again, we can't see the docket. Either way, the case can't drag out and decisions and orders will be rendered in the near future.
It seems pretty certain that unless she produces the children the State will be granted emergency custody. If that happens Lori faces Custodial Interference charges if she doesn't turn them over to the State.
By the way, if she reveals where JJ and Tylee are, the state where they are will take emergency custody of them and either begin a case against Lori or return the children to Idaho. There is no way that she avoids a court battle with a state over the kids. Whether she retains custody in the long term is another matter but there is zero chance that she retains custody in the short term.