You see, this is where I beg to differ and where I almost start agreeing with Cameron and his speech today which upon peeling back the layers is severely limited to what he wishes to achieve by means of current EU rules.
However,
I don't agree that A&E services should be free of charge to non UK citizens.
They aren't even free to UK citizens, just try walking in there and asking for emergency dental treatment and see if you get anything more than a packet of paracetamol and an instruction to ring a dentist, that you'll have to pay for - or more to the point see if you don't get an invoice should you ever be unfortunate enough to need an ambulance after a car wreck.
And if you are taken ill while abroad in the EU you will be entitled to reciprocal basic health care in your EU country but that will vary dramatically depending on which country you are in - try being ill in Menorca for instance and see if you get free treatment, heres a clue, don't try it, just take my word for it, you'll be charged a consultation fee which you then get told to claim back from your insurance.
Whilst I wouldn't suggest anything as vulgar as asking the foreigner under the bus if he has any insurance papers on him, there is plenty of opportunity at every A&E admission to ask the relevant questions, they manage it in all other EU countries and in the USA without it getting in the way of emergency treatment.