WineCountryUA is always a fount of good, useful info, however, I think the following bit is an over-simplification and worth a tweak...

Yes and no. Yes, once moved to United's program, those UR points can only be used to book a ticket through United's MileagePlus program and system. That does NOT mean you can only book a ticket that puts you on a United Airlines plane.

Going through United's MileagePlus system, (right on United's website) you will see (and be able to book) flights on United and also on partner airlines. In your case, this may be moot (a difference without a distinction) for flights from the US mainland to Hawaii (off the top of my head, I can't think of any partner airline that would actually get you from the continental USA to Hawaii), but with United MileagePlus miles, you can indeed book tickets on multiple airlines (just for the record and for future clarification of others) -- at least you could in theory if you were going someplace else.

And now, back to the challenge of finding a reasonable award flight that actually works for your family (a whole other thing)...