Why must the Pythagorean Theorem contain squared values? Is there a relationship between sides prior to squaring them? In a way it is self-evident that the sides of a right triangle have a special relationship to one another; visual proofs illustrate this. However, I am at a loss for how to describe this relationship without squares, and cannot articulate why squares are used other than they make the math simpler.
My intuition tells me that, if the relationship exists after an operation, then it must exist before the operation.  However,  $A^2 + B^2 = C^2$ , but
$A+B ≠ C $
Is my intuition wrong?
I suspect there is a lesson to be learned as to the nature of exponents and their relationship to their roots. 
It seems as though the sides simply exist as square roots a priori.  As if the way they were generated - within the system - defines them as such.
But I can’t describe this process, and I feel as though it would give me a greater intuition into the theory if I did.
Please note that I only have a basic math education.  I have yet to take calculus.  It may well be that the tools needed to articulate an answer require a more advanced math education.  If so I apologize for being inarticulate.  I’m doing the best with the concepts/tools I have available.
 A: If I understood your point, you are questioning why for right triangles we do not have a linear relation among its sides.
Well,  in  a general triangle we have that $$c=a\, cos\beta+ b\, cos\alpha$$ and keeping angles constant, that's a linear relation.
It is to get rid of, or better to find a relation independent of, the angle that in a right triangle you put $c^2=a^2+b^2$. 
A: The squares are required because it's secretly a theorem about area, as illustrated by the picture proofs you've mentioned. Since a side length is a length (obviously), when you square it you get an area. 
The most basic relationship between the lengths of the sides is probably the triangle inequality, and for something specific to right triangles there are any number of trigonometric identities.
A: "Proportional" is a formal mathematical term which describes a linear relationship between two things. We say that $y \propto x$ or $y = kx$ (for some fixed value of $k$) if $x$ and $y$ are proportional to each other. One simple example is how the radius of a circle is proportional to its circumference, because $C = \pi r$ (here, $k=\pi$ and our $x,y$ are the circumference and radius). The term proportional comes from the same root at proportion - a linear relationship is where increasing one thing makes the other change by the same proportion.
If you're still uncomfortable with what "proportional" means, let's take it back a step. If we had for $x = 1, 2, 3, ...$ that $y = 6, 12, 18, ...$, this is a proportional relationship, as (for example) doubling $x$ causes $y$ to double ($y = 6x$ is the exact relationship). But if we had $y = 1, 4, 9, 16, ...$, then there is a connection ($y = x^2$), but doubling $x$ does not double $y$ (it quadruples it), so the relationship is not that the two are proportional.
This term "proportional" is distinct from saying that two things are dependent, or that there is some relationship between them. For instance, the volume of a sphere can be expressed in the equation $V = \frac{4}{3} \pi r^3$. Here, $V$ and $r$ are not proportional, because we need exponents (the $^3$ power) to represent the relationship, but they are related, as given one you can calculate the other. Alternatively, we can say that $V$ and $r^3$ are proportional, because our constant $k$ is $\frac{4}{3} \pi$.
I would surmise that in the case of a triangle, the relationship between the sides requires squaring (raising to the power of two), because the shape is two-dimensional. The circle and sphere examples above also follow this rule: the number of dimensions determines the exponents needed to represent simple relationships like those involving side length, area and volume. Note also that an equilateral triangle's area is given by $A =\frac{\sqrt{3}}{4}x^2$, where the side length is $x$. (Slightly more complicatedly, you may be able to see that the general triangle's area, $A = \frac{1}{2} bh$, follows the same pattern as one variable on the left is determined by the multiplication of two variables on the right.) This may be the intuitive explanation you are looking for.
A: I believe this should help:

The Pythagorean theorem is proven (in the standard proof, the diagram for which is shown here) by relating the area of two squares, one with side length $a+b$ and one with side length $c$. The squares on the terms naturally arise when considering the area of a square.
