I'm currently working on a CoreXY printer, my build platform will be 220 x 220 mm I've chosen to go with linear rails on X and Y axes. I have some nice straight machined 12 mm diameter lead screws on hand which I'm thinking of using in the Z-axis, both of them will be supported on each end with pillow blocks, is it necessary to support them with rails? Or will it be overkill?
Using rails for Z is overkill. They are stiffer than smooth rods, but for Z axis that's not needed.
Using only the leadscrew may be enough, especially in designs with three of them, but if you have only two you may be able to still twist the bed around the Z axis. Also, the bed would be supported only in the center by the "nut" on the leadscrew. This means that the bed could wiggle around the X axis (the axis along the line between the two nuts).
Some additional support for the bed, or using two "nuts" per leadscrew, properly spaced apart, is usually needed.
This in general. Since you mention now that the bed is 220x220 mm, maybe you can get away with the two leadscrews alone, especially if you can find "long" leadscrew nuts, so that the wiggling is not excessive.