Many UK politicians want a deal that would allow the UK to keep key single market advantages while also limiting the numbers of EU migrants entering the UK.
Mr Juncker warned against "secret talks in dark rooms, with curtains drawn, with British government representatives".
"If we start unravelling the [EU] internal market... we will set in train the end of Europe," he said.
There are just under 1.2m UK citizens living in other EU countries, the largest group being in Spain (just under 310,000), followed by Ireland (255,000) and France (185,000), according to United Nations data.
Wynne Edwards, a member of Fair Deal for Expats living in France, said Mr Juncker's "order" was an attempt to influence legitimate discussion on Brexit among UK citizens, and so amounted to discrimination.
"If Germans wanted to speak about the effects of Brexit on Germany they wouldn't be prohibited from doing so," he told the BBC Radio 4 Today programme.
Healthcare is among the key concerns of British expats, he said, citing the case of a Briton who had been getting cancer treatment in France, but had been refused some medication in the UK.
"We ought to be allowed to find out what the lines in the sand are, what is negotiable and what isn't," he said.