Dear David,
You're quite right, article 4 paragraph 2 of the new law says the foreigner, when crossing the border into Slovakia, is required to "submit proof of medical insurance connected with the provision of health care to the foreigner during the stay in Slovakia, if international agreements do not state otherwise."
However, while 17 countries have agreements on the free provision of health care to each other's nationals, this may not mean what you think. For one thing, the 1976 agreement between Slovakia and the UK covers free "medical examination", although you still have to pay the "surcharges normal in each country". It doesn't say anything about complete, free medical care for a serious injury or complicated health condition (see www.health.gov.sk for a list of countries with bilateral agreements and what terms apply).
The other thing is that, according to our research this week, the border police do not have lists of which nationals they are supposed to demand health insurance certificates from and which not. They do, however, have the right to demand it of any foreign national, meaning that if you get a bloody-minded border guard, it may do you little good to insist that the UK-Slovak treaty entitles you to an exemption.
So, rather than misleading, the notice in our paper you referred to was erring on the side of caution, and trying to avoid people getting turned away for the want of a very simple document. The notice in question was also submitted to the paper by the American Chamber of Commerce, which of course was addressing the situation for US citizens who do not, unlike the UK, have a reciprocal health agreement with Slovakia.
I also think you've mixed up two things - visas and border crossings. As a Brit you can stay up to 180 days per calendar year in Slovakia without a visa, although you can't work or attend school. In order to get a student visa, as an EU citizen you should be able to apply directly at a border and aliens police department in Slovakia; everyone else has to apply through a Slovak embassy in their home countries. You can obtain a copy of the procedures at the Slovak embassy in your country, or in Slovak at a police station here.
But as a foreigner you have to obey the border crossing rules like any other foreign national, ie show a valid passport, prove you have about $50 a day to cover the costs of your intended stay, submit to an inspection and - until further notice - show proof of medical coverage.
As soon as the situation clears up and the border police get more precise directions we'll let you know.
Tom Nicholson