BSB and EAP are not common GIS file formats. According to the NV Charts FAQ, the data are intended to be used in the NV Charts app, but they can also be opened in the following programs:
NV Chart Navigator Lite, Standard, Professional; Rosepoint NS;
Stentec; SW & N; Tiki; GPSNavX; Memory-Map; Euronav; Fugawi; GPS 32;
Touratech; expedition; Nautibus; Free Technics; OziExplorer; Navkom;
Barcosoft (Linux); Hatje computer science; Imray; Global Mapper; compegps;
PolarNavy; ConfiTek; OpenCPN; AF tracker; PC-Navigo
QGIS uses the GDAL library to translate geospatial data formats. Here's what it says about BSB files on GDAL's "Supported raster formats" documentation page:
BSB -- Maptech/NOAA BSB Nautical Chart Format
BSB Nautical Chart format is supported for read access, including reading the colour table and the reference points (as GCPs). Note that the .BSB files cannot be selected directly. Instead select the .KAP files. Versions 1.1, 2.0 and 3.0 have been tested successfully.
This driver should also support GEO/NOS format as supplied by Softchart. These files normally have the extension .nos with associated .geo files containing georeferencing ... the .geo files are currently ignored.
This driver is based on work by Mike Higgins. See the frmts/bsb/bsb_read.c files for details on patents affecting BSB format.
Starting with GDAL 1.6.0, it is possible to select an alternate color palette via the BSB_PALETTE configuration option. The default value is RGB. Other common values that can be found are : DAY, DSK, NGT, NGR, GRY, PRC, PRG...
NOTE: Implemented as gdal/frmts/bsb/bsbdataset.cpp.
As you can see from this description, the GDAL definition of BSB files assumes they are the format provided by NOAA, which comes with a KAP file.
EAP files are not mentioned on either the raster or supported vector format pages.
You mentioned that
Loading the .eap data works but there is only black and white lines and nothings looks like a map.
Have a look at the attribute table for hidden information that might be associated with these lines. QGIS allows you to control line styling based on attribute information. Try applying a Categorized, Graduated or Rule-Based style.