In my opinion, not a chance. They are not solo ships, and the show clearly has several crew members running the ship from various stations. So to that end, any starship piloting feats would apply, and you'd need Operations (whatever) skills to make it work. The whole reason the Whitestar was so impressive (well other than jump engines) was that it was capital ship sized (and therefore had a capital ship's compliment of weapons) but could handle nearly like a fighter.
I don't see any reason why humans couldn't pilot a Whitestar, or in fact any other race. But the Minbari control the means of production, so to speak, and it's their technology. Bear in mind that there's a -4 penalty on using alien tech, so the Minbari would be the best at using the Whitestars. The human/Minbari hybrid vessels from Legend of the Rangers probably negate that problem. I'd also house rule it that any Ranger of any race had the tech penalty beaten out of them in training, and could use Minbari technology without penalty, including the Whitestars, but that's just me.
In the last episode, Sheridan takes a ship out solo, but I think that's a "Bluestar", a scaled-down version of a Whitestar (or a scaled up version of a Minbari flyer, take your pick) that is made to be a personal flyer/fighter type craft.