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Square numbers hold a special place in mathematics due to their unique properties. One of the most fundamental properties is that they are always non-negative. A square number is the product of a number multiplied by itself, mathematically represented as \( n^2 \). Because the product of two positive numbers is positive, and the product of two negative numbers is also positive, any real number squared will yield a non-negative result. Additionally, square numbers form a sequence where each term is the square of an integer, starting from zero—0, 1, 4, 9, 16, and so on.

Another key property is that the square of an integer is either a multiple of 4 or one more than a multiple of 4. This is because any integer can be expressed as either 2k or (2k+1), where k is an integer. Squaring these yields either \( 4k^2 \) or \( 4k^2 + 4k + 1 \), explaining the pattern. When dealing with square numbers, their predictability and patterns make them incredibly useful for problem-solving in various fields of mathematics, including algebra, number theory, and calculus.

Calculus, the mathematical study of continuous change, is a pillar of modern science and engineering. One of its primary tools is the function, which relates inputs to outputs in a systematic way. Solving problems in calculus often involves understanding the properties of these functions and how they behave.

For instance, knowing that a function represents a square number can inform us about its graph, its limits, and its continuity. In our exercise, by recognizing that \( (x-y)^2 \) is a square number, we immediately know that the output of the function \( z = x^2 - 2xy + y^2 \) must be non-negative. This is because, calculus-wise, this square function represents the area of a square with side length \( |x-y| \), which cannot be negative.

Moreover, calculus problem-solving often leverages the concept of derivatives, which represents the rate of change of functions. Quadratic functions, a category that includes squared numbers, have well-understood derivatives that further illuminate the nature of their graphs. Ultimately, applying calculus concepts effectively requires a combination of theoretical knowledge and practical problem-solving skills.

The squared difference function is a particular type of function encountered in mathematics, especially calculus and statistical fields. It is generally denoted as \( (a - b)^2 \), representing the square of the difference between two variables or numbers, 'a' and 'b'.

The significance of the squared difference function lies in its ability to always yield a non-negative value, which makes it valuable for measuring the 'distance' between numbers. The non-negativity arises because squaring eradicates the sign of the original difference, whether it was positive or negative. This characteristic of the squared difference function is harnessed in various applications, such as computing the variance and standard deviation in statistics, to quantify dispersion.

Furthermore, when optimized, it can be used to find least squares solutions, which are vital in regression analysis and curve fitting. In our exercise's context, the function \( z = x^2 - 2xy + y^2 \) is a squared difference function with variables x and y. Its graph forms a parabola that opens upward, touching the x-axis at the point (x=y), and never dips below it, reaffirming the non-negative nature of the values of z.