Load the shapefile into QGIS and select WGS84 (EPSG 4326) as the coordinate reference system.
By using a base layer, we can confirm that QGIS has the shapefile in the correct location.
To visualize the map not horizontally stretched, change the CRS of the QGIS project. Using "on-the-fly" reprojection, QGIS will display the map in a different projection without the need to re-project the shapefile. Make sure "on-the-fly" CRS transformation is turned on in the project properties.
For example, here's what it looks with the same project in USA_Contiguous_Albers_Equal_Area_Conic. As the name implies, this projection maintains relatively accurate areas on the scale of the entire contiguous United States. Distances and shapes may be distorted at the expense of keeping area accurate.
This is all you need to do if you just want the map to look less horizontally stretched.
If you need to do calculations, you should re-project the shapefile into an appropriate CRS. Choose a CRS appropriate to the type and scale of calculations you plan to do. Some CRS's are good for area calculations, while others are better for distance calculations. If you're working on a state or county level, choose the UTM CRS specific to that state.
For example, Ohio has these options:
- NAD83 / Ohio North
- NAD83 / Ohio North (ftUS)
- NAD83 / Ohio South
- NAD83 / Ohio South (ftUS)
NOTE: This particular shapefile has Alaska and Hawaii displaced on top of Mexico. Any calculations including those states could be hugely inaccurate. If this is important to how you'll be using this shapefile, you should correct this issue before proceeding. I would recommend posting another question dealing with that issue separately.