Vector data displaying incorrectly

TLDR

The solution is straightforward - use ArcCatalog to reproject the files (both the raster and the vector) to "WGS84" (EPSG 4326) or "Spherical Web Mercator Aux" (EPSG 3857). The screen capture below shows the ArcCatalog output in QGIS 3.12.

Explanation

Firstly, the actual data is in the ArcInfo Binary Grid raster (set of files), and not the Esri shapefile vector (set of files). The vector is mainly a visual guide to show the User where the raster should appear. This is useful because some GIS app, on default, shows mono-band raster values in shades of near white, and against white background, can cause the User to mistaken that nothing was showing, i.e., something was "wrong" with the data, etc.

Secondly, both the raster and vector (set of files) are in Esri's Cylindrical Equal Area projection but (at this time of writing) GDAL/OGR/QGIS (with PROJ6) are having problem identifying it. (See the photo below.) A search on the Internet shows that the PROJ equivalent is +proj=cea. Hence the equivalent PROJ projection string for the above data ought to be:-

+proj=cea +ellps=WGS84 +lon_0=-160.0 +lat_ts=0.0

The GDAL/OGR (with PROJ6) tools have different problems deciphering those files. gdalwarp has difficulties deciphering the negative extent, while ogr2ogr causes some vector points to wrap around the 180W and 180E borders (as you had shown in the screen shot).