Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black.
How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares? 
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.
To solve this problem, we need to determine the number of valid colorings of an \( n \times n \) grid such that every \( 2 \times 2 \) sub-grid contains exactly two white and two black squares. Let's break down the solution step by step.

1. **Understanding the Constraints**:
   - Each \( 2 \times 2 \) sub-grid must have exactly two white and two black squares.
   - This implies that the coloring must follow a specific pattern to satisfy the condition for all \( 2 \times 2 \) sub-grids.

2. **Pattern Analysis**:
   - Consider the smallest grid, a \( 2 \times 2 \) grid. The only valid configurations are:
     \[
     \begin{array}{|c|c|}
     \hline
     W & B \\
     \hline
     B & W \\
     \hline
     \end{array}
     \quad \text{or} \quad
     \begin{array}{|c|c|}
     \hline
     B & W \\
     \hline
     W & B \\
     \hline
     \end{array}
     \]
   - These patterns must repeat throughout the entire \( n \times n \) grid to ensure that every \( 2 \times 2 \) sub-grid meets the requirement.

3. **Extending to Larger Grids**:
   - For an \( n \times n \) grid, the valid configurations must alternate in a checkerboard pattern.
   - There are two possible checkerboard patterns:
     \[
     \begin{array}{|c|c|c|c|}
     \hline
     W & B & W & B \\
     \hline
     B & W & B & W \\
     \hline
     W & B & W & B \\
     \hline
     B & W & B & W \\
     \hline
     \end{array}
     \quad \text{or} \quad
     \begin{array}{|c|c|c|c|}
     \hline
     B & W & B & W \\
     \hline
     W & B & W & B \\
     \hline
     B & W & B & W \\
     \hline
     W & B & W & B \\
     \hline
     \end{array}
     \]

4. **Counting the Configurations**:
   - Since the grid must follow one of the two checkerboard patterns, there are exactly 2 valid colorings for any \( n \times n \) grid.

Conclusion:
The number of valid colorings of an \( n \times n \) grid, where every \( 2 \times 2 \) sub-grid contains exactly two white and two black squares, is 2.

The final answer is \(\boxed{2}\).