Do we ever consider the size of the pixel to calculate the color in Ray Tracing? From what I'm reading, it looks like a ray tracer views a pixel as one color and computes that color using the ray starting in the center of that pixel. Yet a pixel has a width and a height.

Do we ever use the actual square (now a day pixels are considered squares) that the pixel represents to calculate it true color from all possible source within that entire square?

I am thinking that this is how we come up with colors on edges of shapes in scanline rendering and was wondering whether the same could be applied to ray tracing.

There is a picture which I hope better represents what I am trying to ask here. Should we only consider the pixels to be single points? Or are we to consider pixels as large areas?
Yes, and there are several ways this can apply. First off though, we need to fix a misconception in your diagram. Even if pixels really were square, the volume contributing to the pixel gets wider as it gets further away from the image plane, because of perspective. For a square pixel, the volume would be a square pyramid, whose apex is at the camera. It's not quite a regular square pyramid, because the apex is not in the centre of the square: the outer pixels are very skewed. But pixels aren't square anyway, they're cone-shaped.

First off, some people have experimented with cone tracing, where instead of a single ray, you trace a cone with a half-angle determined from the area of the image covered. It's not very popular, because when the cone intersects the edge of a polygon, you need special handling for the remainder of the cone - the part that wasn't blocked by the polygon. For a square pyramid, it's a little easier, because you can change the shape of the base as parts of the pyramid are blocked, but it's still tricky and slow.

You also don't gain much from this technique compared to just supersampling. Even if you trace rays, you don't just trace a single ray through the centre of each pixel. You trace many rays across the area of the pixel, and use a reconstruction filter to take the shape of the pixel into account. This technique has many possibilities that cone tracing doesn't offer: it works better with volumes, you can jitter your samples to avoid structured artefacts, you can use full path-tracing techniques, etc.

Another area where the pixel size matters is in texture sampling. If you want to get nice texturing, you don't just take a point sample of the texture, you set some filter width according to the area of the texture the sample relates to. This doesn't just make your sampling more efficient by using the right mipmap level: it is required for correct sampling to avoid sampling artefacts. In a GPU fragment shader, you might use the dFdx and dFdy functions to do this. In a ray-tracer, you might have use a cone-tracing style of carrying the filter width with the ray, initially based on the level of supersampling at the camera, but updated whenever a child ray is spawned.