The bar is probably intended for a saw with smaller diameter bar studs. The McCulloch 10 Series and 600 Series have 5/16" or 8 mm bar studs, the smaller saws have 1/4" or 6.3 or 6 mm bar studs.
Save that bar and chain for an appropriate saw, and keep looking for a bar that actually fits the 3/8" bar studs on your 250. I would not change the bar studs in any case.
If you absolutely must run that 20" bar on the 250, consider widening the bar slot with a file. Just make sure the oiler and adjuster holes will line up with your 250 before you start messing with the bar. It is also possible to "adjust" the oiler and adjuster holes with a round file provided they are close in the first place.
Most of the time, the 250 would have been equipped with a spur type sprocket and .404 pitch chain. Your new combination is probably 3/8 pitch chain and won't fit the sprocket on the 250 anyway.
Mark