We have been using 0-360 in PostGIS for years but that doesn’t mean it was the most interoperable choice. Using a -180 to 180 (East) system just helps with compatibility across all the different tools and clients. For example, we have always used 0-360 for our nomenclature database but we also support writing out KML files for Google Earth. KML only allows -180 to 180 (East), so during export we swap Longitude range of all points from 180-360 to -180 to 0. I think we leave the Esri shapefile export as 0 to 360 (East) which doesn’t always work well for GISs (like QGIS). This example tutorial in QGIS was more about learning the process, since there there are Python scripts to help swap longitude systems (even in ogr2ogr – the trick is to clip lines and polygons before moving the feature).
But I am surprise the MySQL database is forcing your hand though, it shouldn’t care. That said, we might be playing some games to get this to work – I think by setting the SRID to “-1” (undefined). That means we probably step in-front of any coordinates system transformation and do it our self (for example, projecting from degrees to polar (in meters)). And I know when we export to shapefile, we assign the projection (*.prj WKT projection string) after export.
In summary, if you want things to (usually) just work – use a -180 to 180 system. If you must use a 0 to 360 system, it should work, but be ready to play some games and potentially manually step-in for some use cases. Another method is to use meters (meaning a map projection, where you can set a central meridian to 180).