Why do forecast areas on weather.gov appear as twisted squares?
The equal-area projection used to create the continental U.S. grid for weather forecasting does not represent lines of latitude and longitude as straight lines.  Instead, they are curved (note the Canada-USA border curvature below, which is actually along the 49th latitude parallel).

An equal-area projection is often used in regional weather modeling because it is important for each cell to represent the same amount of physical space during computation.  The map images posted in the question are using the Mercator (or Mercator for web) projection, where lines of latitude are horizontal and lines of longitude are vertical, but area is distorted. 

The grid-cells from the regional model are also designed in a way that maximizes cost-benefit of computational power.  You want to minimize the number of grid cells modeled while maximizing the number of grid-cells in your domain that are relevant to your study area.  This ensures that you aren't wasting computational power and is often achieved by changing the orientation of the domain a little.  So, the grid cells do not align nicely on the mercator map used on web-app maps like google, bing, etc.  However, the grid-cells line up perfectly if visualized using the native projection for the image frame (e.g. below).

The reason that Hawaii does not follow the pattern is that the data for Hawaii is obtained from a model simulation for a different grid.  The nesting of grids can be seen in the image below.

I suggest you read Grids in Numerical Weather and Climate Models
 for a nice look at the factors involved when choosing a grid definition.
You could also look here for more info about NOAA prediction.


North American Mesoscale (NSM) forecast model domains used by NOAA.