In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of  square areas.
1. **Identify the roots of the polynomial:**
   The given polynomial is \( P(x) = x^4 + px^3 + qx^2 + rx + s \). We are told that the roots of this polynomial correspond to the vertices of a square in the complex plane. Let the roots be \( \alpha + \beta i, \alpha - \beta i, -\alpha + \beta i, -\alpha - \beta i \), where \( \alpha \) and \( \beta \) are real numbers and \( \beta \neq 0 \).

2. **Form the polynomial using Vieta's formulas:**
   Using Vieta's formulas, the polynomial with these roots can be written as:
   \[
   P(x) = (x - (\alpha + \beta i))(x - (\alpha - \beta i))(x - (-\alpha + \beta i))(x - (-\alpha - \beta i))
   \]
   Simplifying this product:
   \[
   P(x) = ((x - \alpha - \beta i)(x - \alpha + \beta i))((x + \alpha - \beta i)(x + \alpha + \beta i))
   \]
   \[
   = ((x - \alpha)^2 - (\beta i)^2)((x + \alpha)^2 - (\beta i)^2)
   \]
   \[
   = ((x - \alpha)^2 + \beta^2)((x + \alpha)^2 + \beta^2)
   \]
   \[
   = (x^2 - 2\alpha x + \alpha^2 + \beta^2)(x^2 + 2\alpha x + \alpha^2 + \beta^2)
   \]
   \[
   = (x^2 + \alpha^2 + \beta^2 - 2\alpha x)(x^2 + \alpha^2 + \beta^2 + 2\alpha x)
   \]
   \[
   = (x^2 + \alpha^2 + \beta^2)^2 - (2\alpha x)^2
   \]
   \[
   = x^4 + 2(\alpha^2 + \beta^2)x^2 + (\alpha^2 + \beta^2)^2 - 4\alpha^2 x^2
   \]
   \[
   = x^4 + (\alpha^2 + \beta^2)x^2 + (\alpha^2 + \beta^2)^2
   \]

3. **Determine the integer coefficients:**
   For the polynomial to have integer coefficients, the terms involving \(\alpha\) and \(\beta\) must be integers. Specifically, \(\alpha^2 + \beta^2\) must be an integer.

4. **Minimize the area of the square:**
   The area of the square is given by the square of the side length. The side length of the square is \(2|\beta|\). Therefore, the area of the square is:
   \[
   \text{Area} = (2|\beta|)^2 = 4\beta^2
   \]
   To minimize the area, we need to minimize \(\beta^2\). Since \(\alpha\) and \(\beta\) are integers, the smallest non-zero value for \(\beta\) is 1.

5. **Conclusion:**
   The minimum value of \(\beta\) is 1, which gives the minimum area of the square as:
   \[
   4 \times 1^2 = 4
   \]

The final answer is \(\boxed{4}\).