Is there a way to change the coordinates of a shapefile (points) from lat/long to UTM or Gauß-Krüger within the same CRS (i.e. WGS84:4326)?
In general, you can reproject any shapefile to your desired CRS/Projection by just saving a new copy of it with that CRS applied (layer -> right click -> save as... and set desired CRS in dialog).
But what do you mean by 'within the same CRS'? UTM projection uses WGS84 or GRS80 as reference elipsoid, Gauss-Krüger however is projected and bound on/to either the Bessel or Krassowski elipsoid. You can reproject your lat/long to both, but for Gauss-Krüger you cannot (and don´t need to) keep the WGS84 reference system.