The key is that the overnight (less than 24 hours) has to be en-route, and don't call it a "stopover".
Now, to try to answer your question, on an award ticket you can get one stopover (defined as a scheduled layover > 24 hours on an international itinerary) with some geographical restrictions on one award ticket. So the OP has an award ticket to Thailand (South Asia). He's taking his one stopover in Osaka, Japan on the way to Thailand. So he stays in Osaka for a week (or however long he wants to, subject to the rules of the award ticket) and stays at his destination in Thailand for however long he wants to. He meets the geographical restrictions because Japan and Thailand are in different geographical regions as defined by the award schedule.
If he can manage it, he can also overnight in Bangkok (for less than 24 hours) on the way to and from Phuket. If you're flying in or out of Bangkok on UA, this is easy (in fact it's required) to do because the UA NRT-BKK flight arrives very late (too late to catch a domestic flight to HKT) and the UA BKK-NRT flight leaves very early (too early to make it from HKT the same day), so you must overnight in BKK in both these cases. The overnights must be arranged so they are less than 24 hours between scheduled arrival and scheduled departure times, in which case they are not considered stopovers on an international itinerary.
Your problem with trying to get a stopover is that your award ticket is to Japan, which is one of the geographical regions defined by the award charts. UA flies directly to Japan, so there's no place to stopover in Japan. You might be able to do a stopover in Honolulu on the way out or the way back on that award. However, if you want to also go to SIN, BKK or HKG, you need to trade in your award for a South Asia award. Then you can do the same thing the OP is doing. Take a stopover in Japan (geo region = Japan) on the way out or the way back and also visit your single destination city in (what the awards chart calls) the South Asia geographical region.
-David