First: the EU is not a judicial unity, and the power of the "EU" (or its organs) does not extend to the judicial system (cf. the fight with Poland).
Second: All jurisdictions in the EU have "ne bis in idem" (which is part of the 5th, right?), and in the ones I know of, the suspect/defendant cannot be made to reply. Where I live, e.g. passwords are also under this doctrine, but not under all conditions. The 4th amendment is common too, although there are concerns that some member states aren't too concerned with a suspect's rights.
Australia's problems don't automatically apply to the EU, not even from a USA-centric point of view.
