Really, this isn't as implausible as some people are making it out to be. The caterpillar track, the basis for an armoured, tracked vehicle, was designed, albeit crudely, in 1770, almost a century before the ACW. Mikey is right, also, in that the British did use some large, steam-driven tractors using the caterpillar track, to manoeuvre on the Crimean War's muddy battlefields. The potential is there, but no one realized it. If we have some enlightened military strategist (or anyone, really) realize the potential for the vehicle, I see no reason that steam-powered tanks (probably using some sort of Gatling gun rather than the cannon we have in OTL, although that could come later.) couldn't duke it out in Union and Confederate forces.
Granted, steam powered tanks, in full service this early, would be blazing hot, claustrophobic nightmares, but that wouldn't be too different than the OTL tanks of WW1. I'm sure many soldiers would take armoured protection over comfort in the breeze.
Plus, it wouldn't be too hard to speed up the invention of the internal combustion engine, which, if I remember correctly, was invented in 1885, only twenty years after the ACW's end. If we could get an efficient gas powered engine during the Crimean War, tanks by the ACW gets a bit more plausible.