A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.
To solve this problem, we need to ensure that each \(1 \times 1 \times 4\) rectangular box contains exactly one red unit cube. This means that in each column of the \(4 \times 4 \times 4\) cube, there must be exactly one red unit cube. This is equivalent to finding a Latin square for each layer of the cube.

1. **Understanding the Problem:**
   - We have a \(4 \times 4 \times 4\) cube composed of 64 unit cubes.
   - We need to color exactly 16 unit cubes red such that each \(1 \times 1 \times 4\) column contains exactly one red unit cube.

2. **Latin Square Configuration:**
   - A Latin square of order 4 is a \(4 \times 4\) array filled with numbers 1 to 4, each occurring exactly once in each row and column.
   - We need to find all possible Latin squares for the \(4 \times 4\) layers of the cube.

3. **Constructing the Latin Squares:**
   - Start with an initial configuration:
     \[
     \begin{array}{cccc}
     1 & 0 & 0 & 0 \\
     2 & 1 & 0 & 0 \\
     3 & 0 & 1 & 0 \\
     4 & 0 & 0 & 1 \\
     \end{array}
     \]
   - Fill in the zeros to complete the Latin square. We can generate different Latin squares by permuting the rows and columns.

4. **Generating All Latin Squares:**
   - The number of distinct Latin squares of order 4 is 576. This can be calculated as \(4! \times 3! = 24 \times 6 = 144\), but since we need to consider all possible permutations of the layers, we multiply by 4 (for each layer):
     \[
     4 \times 4! \times 3! = 4 \times 24 \times 6 = 576
     \]

5. **Conclusion:**
   - The number of interesting colorings is 576.

The final answer is \(\boxed{576}\).