There are several things that can cause "squares" or squared-off edges in your picture. It seems that you might be experiencing several different kinds:
"...screens that make a whole image devolve into a series of squares."
This is most probably caused by a momentary loss of signal (LOS) in the video stream somewhere between the source and your TV. Digital TV pictures distort differently from old analog ones, and the picture disintegrating into large square areas is a common form of distortion. Non-technical people have latched onto improper terminology that's wrong, but the names have stuck. So if you call it "pixelization" or "macroblocking", it would mean something to a teenager working at the local Best Buy store. If this is a persistent problem that needs solving, you'll need more skilled people to troubleshoot and remedy it, though.
If the squares are small, and all over the screen, but come and go quickly during fast action scenes, this is an artifact of the compression algorithm that is used to save bandwidth. Currently used algorithms can't handle motion well, and seeing the picture degrade into a sort of mosaic pattern is a common artifact of this. Once again, terms like "motion blur" are often used improperly to describe these artifacts that, if anything look overly sharp.
"The natural curve of a person's face is seen in squares that makes the outline appear in small, linear units."
This is a different phenomenon known as quantization error, or "jaggies". The word "aliasing" is often used (and misused) to talk about this. The most common cause of jaggies in TV is when older standard definition images are scaled up to HD resolution without extra signal processing that can smooth out the jaggies. It can happen whenever there's a difference in the number of raster scan lines between two different TV systems. Not much you can do about this, as the time and place to address it is during the conversion process.
Note that jaggies and "motion blur" [sic] can look quite similar to the untrained eye.
"In newspaper images, I believe that the series of "dots," is referred to as the matrix,, but I don't know if that translates to a broadcast/aired picture."
It's the same basic theory, although with TV images each row of dots is cut off from the whole picture and serialized into a stream of dots which are then transmitted or stored. The stream is reassembled on the screen (not unlike how a shredded document can be reassembled) into a 2-D image.
Imagine a brick wall, and a painter who paints each brick, one by one, going left to right. When the painter gets to the last column of bricks, he goes back to the left side, and repeats the same process for the next lower row of bricks, brick by brick, line by line until the whole wall is painted. When the last brick at the bottom right is finished, the painter goes up to the top left brick and starts again... This is how television has worked, even before digital. In TV we call each "dot" a "picture element", or "pixel" for short. Once again here's a word that's often misunderstood and misused!