Derp - forgot an important point:
American bikes used thick tubing in small diameters (and often Ashtabula's thick, heavy forks,) while European and Japanese bikes used thin tubing in large diameters. Over time everyone has mostly shifted over to European-style tubing, which is why a BMX 7/8" (thin tube*, thin steer tube) and a Schwinn 1" (thick headset tube, thick steer tube) use the same stem. However, a 1" BMX headset will probably fit your bike.

*or extremely thick tube with the same inner diameter for making "sweet jumps"

More confusion: Schwinn used European parts on their high-end bikes, and by the 70s they were selling Japanese-built bikes as their own. If you go the eBay route, be aware that a Schwinn quill may or may not be the right size for your bike.
While some think "ABS" means a part was made by Ashtabula, it really means "anti-lock brake system." If you want to do some sw33t skidz, you should remove those parts immediately and throw them in the trash. Especially Stingray forks.