Secession to the Confederacy perhaps, but not to become their own country- especially when the war to settle that issue just ended five minutes earlier in a Union victory. That massive army that was formed to end the Confederacy just gets turned around and pointed in the general direction of the Pacific Ocean for either Californian separation or British takeover.California was actually close to secession.
I will concede to this point in part. The massive influx of Americans would have been stilted by a continued war effort. Britain did own much of the land at points, though yes, it was joint occupation at the time. As for the take over, it would have been mostly politically fueled. No one would want to continue fighting after that, especially to regain what was at the point largely a Territory. Granted Oregon was a state at the time. As for the name, no, it probably would not have been changed, though the name does exist and was used for the Cascade Rapids.
The massive influx was there before the War- and even with the War, population in these regions doubled between the 1860 and 1870 censuses (censii?). Washington and Oregon between them had 50,000 inhabitants in 1860; BC had 11,000 (and a large number of these were expat Americans looking for gold). The United States isn't going to sit aside and let the British walk all over them when just five years earlier the British knew that the Americans would have gone to war in Canada over the Trent Affair, and would have fought that war with the leftovers from the fight in the South. If Britain goes in to the Northwest at this point, Canada burns.
(And the name wouldn't be Cascadia not because the name didn't exist, but because no one ever called the greater Northwest 'Cascadia' until well into the 20th century. Oregon Country, the Columbia District, New Albion if you look really far back, but 'Cascadia' is a modern phenomenon.)