You're using the wrong tools; using EF to search AA Milesavers won't find you anything from HKG, BKK or SIN.
If you want to search online, you will need to use CX's own award search tool (which requires a free membership of Asia Miles.) Awards using AA miles on CX match what is available on CX website most of the time, though some people have reported discrepancies.
Otherwise, if you are using AA miles you should call AA to check availability and book. Be prepared to offer alternative routes and dates.

Having done a quick search on CX, it seems that HKG-JFK is available in J on Dec 26 (departing at 01.00 arr. JFK at 06.10) but not Dec 24 or 25.

HKG-ORD is available Dec 24 and 25 (departing 11.35 arr. ORD at 12.40) from where you should be able to find an AA connection to JFK/LGA.

BKK-HKG is available on all four flights on Dec 24, but none get to HKG in time for 11.35 ORD flight, so you will need to overnight in HKG.

On Dec 25, only available flight from BKK-HKG is at 08.25 arr. HKG at 12.10 so you would have 12 hours in HKG if you go for the 01.00 JFK departure on DEC 26.

I did not check other routes via LAX or YYZ but those may also be available.

Good luck and Happy Travels.