I guess, this might be due to the enclosure chipset. I am not really sure, but I think the crossflash worked with an enclosure based on Cypress AT2 chipset.
You want me to awake my own 1660P1 from its storage location and see if it is recognised by BQflasher when connected to a Jmicron based adapter?
BQFlasher does see my Philips 1660P1, connected to a Jmicron based IDE/USB adapter. VID=0x152D, PID=0x2338. JM20337 is printed onto the chip, as far as I can see.
I have not attemped to flash the drive now.