1. Assume the top left corner is black: This assumption simplifies the problem without loss of generality because the coloring pattern is symmetric.

2. Define variables for the lengths of black parts: Let $x$ be the sum of the lengths of the black parts along the top edge of the square, and let $y$ be the sum of the lengths of the black parts along the left edge of the square.

3. Calculate the area of the black parts: The area of the black parts can be calculated by considering the contributions from the black rectangles. The black rectangles along the top edge contribute $x \cdot y$ to the area, and the black rectangles along the bottom edge contribute $(2-x) \cdot (2-y)$ to the area. Therefore, the total area of the black parts is:
$$\text{Area}_{\text{black}} = xy + (2-x)(2-y)$$

4. Simplify the area expression: Simplify the expression for the total area of the black parts:
$$\text{Area}_{\text{black}} = xy + (2-x)(2-y) = xy + 4 - 2x - 2y + xy = 2xy + 4 - 2x - 2y$$

5. Set the area of black parts equal to 2: Since the total area of the square is 4 and the areas of the black and white parts are equal, the area of the black parts must be 2:
$$2xy + 4 - 2x - 2y = 2$$

6. **Solve for $x$ and $y$**: Rearrange the equation to isolate the terms involving $x$ and $y$:
$$2xy + 4 - 2x - 2y = 2 \implies 2xy - 2x - 2y + 4 = 2 \implies 2xy - 2x - 2y + 2 = 0 \implies 2(xy - x - y + 1) = 0 \implies (x-1)(y-1) = 0$$

7. **Determine the possible values for $x$ and $y$**: The equation $(x-1)(y-1) = 0$ implies that either $x = 1$ or $y = 1$.

8. Reassemble the black rectangles:
- If $x = 1$: This means the black parts along the top edge sum to 1. We can cut out the black rectangles and push them horizontally until they touch the next black rectangle, forming horizontal strips of width 1. The heights of these strips sum to 2, forming a $1 \times 2$ rectangle.
- If $y = 1$: This means the black parts along the left edge sum to 1. We can cut out the black rectangles and push them vertically until they touch the next black rectangle, forming vertical strips of width 1. The heights of these strips sum to 2, forming a $1 \times 2$ rectangle. We can then rotate this rectangle by $90^\circ$ to match the required orientation.

$\blacksquare$