This sort of thing can read like magic unless you're familiar with Maths and there's more to it than just the sign. I could ask you why not just use the magnitude of a value instead of using the square of the value.That is because not direction of velocity but magnitude of velocity matters. v2 has no information of direction. Same for c2.
Squares often come into equations and formulae when there are two quantities multiplied together and one quantity is also due to two multiplied quantities.
So, velocity times time is distance (vt) and velocity is acceleration times the time it's applied (at). This means the distance travelled, after a time t will be the average velocity times time (vaveraget). Starting from 0, the average velocity will be v/2 so the distance travelled will be vt2/2.
Squared quantities often come out of the area of a graph.