Assuming this Speedster is a 26"...... Is it a middleweight? Lightweight? Camelback?
Porkchop BMX had forks (Tange Cr-Mo 26") for a while, but I see they are out. The trouble I see with the one you linked is the specs aren't complete. What I don't see is the steerer tube length. Yes, you will have to cut it. When Porkchop had the Tange forks, they had 2 lengths so you could pick the one you needed, cut it and still have threads. Amazon, as usual, is missing important information. Worst case though, with the Amazon fork, you will have to take it to a bike shop and have it threaded and slotted. I would absolutely plan on this from what I see. Your Schwinn headset will fit, except for possibly the crown race, as there are 2 standards for crown race size on a 1" steerer. This isn't really a problem either. Many aftermarket threaded headsets, good ones anyway, like several models of Tange, use a Schwinn VSF bearing like your Schwinn headset does. You can probably get an appropriate crown race pretty easy if yours is wrong. It might need replacing anyway. They often do. EDIT: Amazon says its a 27mm crown race. I think you'll need a different one.
Measure carefully and don't cut the fork too short! Mock it up with the headset you plan to use (they vary a lot), gonna have a reflector bracket? Cable stop for centerpull or cantilever brakes? You can probably fix too long with a spacer or a Wald top nut, but too short is too short.
Your current stem probably wont fit if the bike is newer than 65 or so. Schwinn beefed up the steerer tube and that made the stem smaller. The fork maker should tell what stem size it takes, but I'll bet Amazon didn't list it. Come to think of it you might need a new nut for the top of the headset if the new fork takes a bigger stem. EDIT: A commenter says 21.1mm. The post 65 (or so) Schwinn stem is probably what fits.
Someone in the comments says the over locknut distance is "98mm", and again Amazon doesn't say. It is probably 100mm, because basically everything is today. Good if so. 100mm is standard now. It is sort of critical on this style fork. Old French hubs are about 97-98mm, are common, and Schwinn used a bunch of them. Depending on how old the Speedster is, you might already have one if it turns out you need 97mm. Or you might have 91mm and need different hub.
There are advantages to threadless, but there's no way I would convert a retro bile like this unless there was some particular stem/bars/fork/whatever I wanted to use that required it. Quill stems are mickey mouse I'll admit, but worked fine for everyone for decades, even in BMX, and the only common problem with them is getting horribly stuck in the fork when someone left the bike outside to rust.