This is especially important when it comes to Ireland.
Both Britain and the European Union have vowed to avoid the return of customs checks to the border and an interim deal in December left some flexibility on the issue, but an European Union text putting the agreement into law has sparked a fresh row with London.
Mr. Juncker says both sides have agreed that there should be no hard border on the island of Ireland and that must be respected.
Juncker told MEPs the draft text simply translated the December accord into legal language and "should not come as a shock".
During his speech, Juncker said the issue of Ireland was a European one, but was interrupted by members of the parliament saying this was only a British issue.
"One can not have at the same time the status of a third country and demand at the same time the advantages of the (European) Union", Barnier told a plenary session of the European Parliament in Strasbourg.
EU Council President Donald Tusk warned last week that the Irish border issue must be solved before negotiations can move on to other issues.
Barnier also made it clear that if Britain wanted to remain in some European Union agencies like those on chemicals, aviation or medicines, it would also have to recognize the authority of the EU's top court in those areas.
Britain has said it no longer wants to be part of the EU's seamless and tariff-free internal market and the customs union.
The agreement would be based around four "pillars" - trade, foreign policy and security cooperation, internal security and "thematic" cooperation, which includes education and research programmes. "That can not happen", the European Parliament's Brexit negotiator Guy Verhofstadt told Euronews.
In a blunt response to May's speech earlier this month, in which she set out her post-Brexit vision, Barnier said the prime minister still didn't get that the European Union wouldn't accept her having the best parts of European Union membership if she didn't play by the bloc's rules.
"We are open for business".
Mrs May's spokesman said Mr Trump had said the United States was "with the UK all the way".