Remember that you need to first extract the styles or swatches from the zip file before you can import them. You can't import the zip file!
You import both Affinity Designer swatches and styles in pretty much the same way. You click on the hamburger menu (a hungry person's hamburger) at the top right of the swatch or styles window and then select the import option. For swatches I import it as an application palette. Then you just pick the file to import and you're done.
If you make any changes to a set of styles or swatches then just use the export option which just like performing a save as.