Afaik, Denmark doesn't require the EHIC for acute emergency care. Anyone is eligible for unbilled emergency care, with some exceptions to combat medical tourism (if you're found to have traveled to Denmark for the specific purpose of entering an ER to treat a health condition). The EHIC qualifies you for some additional care, e.g. childbirth. This is actually a big current sore point in Danish politics, because many other EU countries don't extend a reciprocal courtesy.
Even Canada? Canada by its location would be at acute risk for medical tourism from the US, if it didn't bill for ER treatment. That problem isn't as big for Europe, Australia, etc.
I could almost imagine a financially manageable system where Canada gave free ER care to anyone except Americans, but that would produce obvious political problems.
Also, unless my memory is totally off, each country has to pay up for EHIC, depending on who used what, and visa versa. So the cost should be evened out.
Edit, because I cannot reply to you later down: I think we have a different view on what emergency medical care is. In most of the world you will get fixed up so that you are no longer in life threatening condition, this is the case in USA, Denmark and Norway. But afterwards you will have to pay, even in Denmark, as long as you're not apart of EU and have the European Health Insurance Card or a citizen of any of the Nordic countries, which has a special agreement.
[1] http://www.sos.dk/en/Corporate/OurNetwork/ForMedicalFaciliti...
In the UK you are not, purely because they don't do the accounting that way in the first place as it would cost a lot of extra money to keep track of billing when no-one is being individually billed, so there is no bill to bill.
My travel insurance reimbursed everything but you'd better have money on your credit card...
Before getting my work holiday visa, I had to prove that I had medical insurance covering me during my stay in Canada.
[edit: I'm from Belgium]