1. Define the coordinates and properties of the square:
- Let the center $M$ of the square be the origin $(0,0)$.
- The diagonals of the square have length 2, so the side length of the square is $\sqrt{2}$.
- The vertices of the square $A, B, C, D$ can be represented as:
$$A = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), \quad B = \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), \quad C = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right), \quad D = \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$$

2. **Define the line $\ell$ and the projections:**
- Let $\ell$ be a line at a distance $d$ from the origin $M$, where $d > 1$.
- The orthogonal projections of $A, B, C, D$ onto $\ell$ are $A', B', C', D'$.

3. **Calculate the distance from each vertex to the line $\ell$:**
- Let the equation of the line $\ell$ be $ax + by + c = 0$.
- The distance from a point $(x_1, y_1)$ to the line $ax + by + c = 0$ is given by:
$$\text{Distance} = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$
- For each vertex $A, B, C, D$, the distances to the line $\ell$ are:
$$\text{Distance from } A = \frac{|a\frac{\sqrt{2}}{2} + b\frac{\sqrt{2}}{2} + c|}{\sqrt{a^2 + b^2}}$$
$$\text{Distance from } B = \frac{|a(-\frac{\sqrt{2}}{2}) + b\frac{\sqrt{2}}{2} + c|}{\sqrt{a^2 + b^2}}$$
$$\text{Distance from } C = \frac{|a(-\frac{\sqrt{2}}{2}) + b(-\frac{\sqrt{2}}{2}) + c|}{\sqrt{a^2 + b^2}}$$
$$\text{Distance from } D = \frac{|a\frac{\sqrt{2}}{2} + b(-\frac{\sqrt{2}}{2}) + c|}{\sqrt{a^2 + b^2}}$$

4. Sum of the squares of the distances:
- Let $k = \frac{|c|}{\sqrt{a^2 + b^2}}$, which is the distance from the origin to the line $\ell$.
- The sum of the squares of the distances from the vertices to the line $\ell$ is:
$$AA'^2 + BB'^2 + CC'^2 + DD'^2 = \left(\frac{|a\frac{\sqrt{2}}{2} + b\frac{\sqrt{2}}{2} + c|}{\sqrt{a^2 + b^2}}\right)^2 + \left(\frac{|a(-\frac{\sqrt{2}}{2}) + b\frac{\sqrt{2}}{2} + c|}{\sqrt{a^2 + b^2}}\right)^2 + \left(\frac{|a(-\frac{\sqrt{2}}{2}) + b(-\frac{\sqrt{2}}{2}) + c|}{\sqrt{a^2 + b^2}}\right)^2 + \left(\frac{|a\frac{\sqrt{2}}{2} + b(-\frac{\sqrt{2}}{2}) + c|}{\sqrt{a^2 + b^2}}\right)^2$$

5. Simplify the expression:
- Notice that the terms involving $a$ and $b$ will cancel out due to symmetry, leaving:
$$AA'^2 + BB'^2 + CC'^2 + DD'^2 = 4k^2 + 2\left(\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2\right)$$
- Simplify further:
$$AA'^2 + BB'^2 + CC'^2 + DD'^2 = 4k^2 + 2\left(\frac{1}{2} + \frac{1}{2}\right) = 4k^2 + 2$$

6. Conclusion:
- Since $k$ is the distance from the origin to the line $\ell$, which is fixed, and the term $2$ is constant, the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change when the square is rotated.

$\blacksquare$