Yes, you should be able to stay airside once you arrive in SIN. But note that United flights go through an additional security screening that is normally done before check in on ground side; this screening is by a uniformed officer checking your name off a list to make sure you're on the HKG flight, plus the officer sticks a colored dot on the back of your passport. At the gate, another officer doing the primary screening will see you don't have the dot, not sure what happens then.

I'd suggest that after your nap airside, go to the Transit Desk (it's also airside) and let them know you're there and already have your boarding pass. BTW you need to ask which Transit Desk to go to. UA departs from T2, yet the Changi Airport website says go to the T3 transit desk, which strikes me as being incorrect.

Changi airside is a good place for a nap. I've seen recliners with leg rests, sorry I can't tell you where they are - I live in Singapore so I never had to use them!