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Square taper was only a thing in bmx pretty early on. Much more common is a splined interface with or without pinch bolts. (48 spline without, 8 or thereabouts spline with). Their are other systems, Primo cranks use a non tapered square axle with pinch bolts and Odyssey have a hexagonal/octagonal axle with a complex wedge thing.
Bearing wise though, they are generally a press fit with the bearings either going into cups first then into the frame (USA) or straight into the frame (Spanish or Mid). Mid uses the same bearing as USA without the cup, Spanish is a different bearing.
They have a tube spacer between the bearings which the inner race of the bearing suits against, like the shoulders on a hub axle. Without the tube you could over do the preload. I think it has to be a tube (as opposed to shoulders on the spindle like a hub axle) so you can press the bearings into the frame without having the spindle in the way.
Can you explain BMX bottom brackets , are they just like square taper ones but with a removable axle? Im messing around with one tonight