We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$  how many such colorations are possible?
1. **Understanding the Problem:**
   We need to color an \( n \times n \) board such that any \( 2 \times 2 \) subsquare has exactly two red squares and two black squares. This implies that the board must be colored in a checkerboard pattern.

2. **Checkerboard Pattern:**
   A checkerboard pattern alternates colors in such a way that no two adjacent squares (horizontally or vertically) have the same color. For an \( n \times n \) board, this pattern can be achieved by:
   - Coloring the square at \((i, j)\) red if \(i + j\) is even.
   - Coloring the square at \((i, j)\) black if \(i + j\) is odd.

3. **Uniqueness of the Pattern:**
   To ensure that every \( 2 \times 2 \) subsquare has exactly two red and two black squares, the checkerboard pattern is the only possible solution. Any deviation from this pattern would result in a \( 2 \times 2 \) subsquare that does not meet the requirement.

4. **Counting the Number of Valid Colorations:**
   Since the checkerboard pattern is the only valid pattern that satisfies the given condition, there is exactly one way to color the \( n \times n \) board.

Conclusion:
The number of such colorations is \( \boxed{1} \).