Color $27$ unit cube so that by rearranging, they could form a blue $3\times3$ cube, a green one, and a red one?

For the $3 \times 3 \times 3$ case it is possible:

Haskell code:

{-# LANGUAGE FlexibleContexts #-}

import Diagrams.Prelude
import Diagrams.Backend.Cairo.CmdLine (defaultMain)

v x = [x,x,x]
e x = [x,x]
f x = [x]

cubes =
  [ v red ++ v green
  , v green ++ v blue
  , v blue ++ v red
  ]
  ++
  concatMap (replicate 6)
    [ v red ++ e green ++ f blue
    , v green ++ e blue ++ f red
    , v blue ++ e red ++ f green
    ]
  ++
  replicate 6 ( e red ++ e green ++ e blue )

draw [a,b,c,d,e,f] = pad 1.1 . centerXY $
  ((strutX 1 ||| square 1 # fc a)
  ===
  (square 1 # fc b ||| square 1 # fc c ||| square 1 # fc d ||| square 1 # fc e)
  ===
  (strutX 3 ||| square 1 # fc f))

chunk _ [] = []
chunk n xs = let (ys, zs) = splitAt n xs in ys : chunk n zs

diagram = bg white . centerXY . vcat . map hcat . chunk 4 . map draw $ cubes

main = defaultMain diagram