There isn't any particular difficulty in the characterization of c^2 and it's meaning. The only difficulty is in your understanding.
We've given you a very simple analogy - a foot is different than a foot^2.
For more advanced understanding, we've given you links to the theory of dimensional analysis, which addresses the topic of whether c is different than c^2 more precesisly than the simple, easy-to-understand analogy does or can.
The analogy alone should be enough to at least make you think about why you assume that c is equivalent to c^2. Given that a foot is not equivalent to a foot^2, why should you asusme that a velocity (c) is equivalent to a velocity^2 (c^2)?
The one thing we haven't done (yet) is to spoon-feed you some of the elements of dimensional analysis. I'll try that in a bit, but I do get the feeling that you aren't really hear to learn stuff, you don't seem to be listening very much. Rather than listening, you seem to be making a bunch of more or less unfounded statements, and then attempting to defend them. Anything that doesn't agree with your unfounded claims seems to get mostly ignored.
Before I start, I'm going to ramble on a bit about the role of mathematics in physics. Mathematics is not a hinderance, as you seem to think. It is an esesential tool. Mathematics does not cause errors in understanding. Mathematics greatly helps to eliminate errors. It *is* possible for errors to creep in in spite of mathematics. This happens when one makes incorrect assumptions. Mathematics is a codified form of logic, so it helps to insure that the conclusiosn follow from the premises. It can't necessarily find errors in the fundamental assumptions. It can greatly aid in ensuring that the conclusions follow from the premises.
Enough about mathematics, let's go back to spoon-feeding you some dimensional analysis.
The idea of dimensional analysis is that every phhysical quantity contains two parts: a number, which gives the magnitude of the quantity, and a unit, which describes how the quantity transforms under scale changes.
Scale changes are when one uses different units - like feet, instead of inches, or seconds instead of minutes.
So let's go back to feet and feet^2. There are 12 inches in a foot, so the rules of dimensional analysis tell us that if we have one foot, and we transform it so that it's units are in inches, we get 12 inches. These two expressions represent the same physical quantity, i.e. 1 foot is 12 inches.
The same rules tell us that if we have one square foot, when we transform to inches we get 144 square inches.
You can check this out for yourself if you really want to - take a square foot, and see how many square inches are in it.
Note that the rules of transformation are totally different for square feet than they are for feet.
This is why we say that feet^2 are different than feet. It also means that we can't directly compare quantites in feet and quantities in square feet in any meaningful way. Because the quanties transform differently, the result of comparing the number part of the quantites will not give the same result in different units (remember, every physical quantity has two parts - a number, and a unit).
These rules can be written down very concisely by treating units as quantites which 'cancel out' in fractions.
See for instance
http://www.chemistrycoach.com/use.htm
<br />
1\, foot * \frac{12\, inches}{1\, foot} = 12 inches<br />
Feet appear in the numerator and denominator once, and "cancel out", leaving inches.
<br />
1\, foot^2 *\left( \frac{12\, inches}{1\, foot} \right)*\left( \frac{12\, inches}{1\, foot}\right) = 144 inches^2<br />
Feet appear twice in the numerator and denominator. Both feet "cancel out", leaving inches^2.
Now, let's apply this to velocity.
Suppose we have a velocity of 1 foot/second. The rules of dimensional analysis say that this transforms to 12 inches/second.
Now let's say we have a velocity^2 of 1 foot^2 / second^2. The rules of dimensional anaysis say that this transforms to 144 inches^2/second^2.
Now you can see why a velocity^2 is different than a velocity. The numerical value transforms in a completely different manner when we change units (i.e feet to inches, in this example).
We can also use dimensional analysis to transform the seconds into minutes
60 feet/minute == 1 foot/second
3600 feet^2/minute^2 = 1 foot^2/second^2.
Knowing how to transform both the "feet" (distance), and the "seconds" (time) in the velocity gives us all the information we need to transform a velocity from feet/seconds to any other units we desire. (Furlongs per fortnight, for an extreme example).
This is dimensional analysis in a nutshell. To recap, a physical quantity consists of two parts: a number, AND a unit. Two quantites can be compared directly only if they have the same units. c and c^2 do not have the same units, so they cannot be compared dirrectly.