No, this is simply not true.
In France, the EU is viewed with even wider skepticism than in the UK.
Just before Brexit - the majority of France, Germany, UK and Spain had negative views of the EU [1]
Popular sentiment towards the EU has been declining steadily until Brexit, during which it's shot up somewhat, as EU-wide press does their best to inject fear into the situation - but without this existential angst, the trend-line is clearly going down.
This is also obvious from recent Eurosceptic political trends.
Consider that just this week, the far right AfD in Germany, has just announced they would like to move Germany out of the EU as a material policy. They have about 15% of the vote. They've always been sceptic, but there has never been discussion in Germany about exit before.
But it is of course nuanced:
+ Wether or not citizens want to 'leave' is a different question.
+ Wether or not people would opt to 'join' were they not members otherwise, is also a different question.
+ Wether or not they can separate the notion of 'EU' and 'Europe' is another question. For example, Italy, Europe's 3rd largest economy has just elected a Eurosceptic party, but they have no real inclination to leave. For Italians, to 'leave the EU' is to 'leave Europe' and because they see themselves as European, they may not chose to leave the EU. (I see that as a false equivalence, that the EU is not Europe, just a body politic and set of treaties.)
+ Nations that have the most direct battles with the EU, for example Poland, have the highest rating!
Make no mistake - though material 'exit' may be stronger in the UK than elsewhere, anti-EU sentiment is strong across Europe, and the overall trend-line is bad news.
Aside from political issues, there are serious economic concerns about the Euro, and that it's a 'trap' for peripheral nations unless there is deeper political integration (there won't be), and even if there was, they would never agree on how to print money anyhow.
And finally - even issues for which the EU can't fairly be blamed ... they will take the heat for in many cases.
The kind of populism you see in France right now with the Gilets Jaunes which is leaderless, has no ideology, there's nobody to negotiate with, nobody to appease, nobody really knows for sure why it's happening, it's just 'angst' ... this is going to spill over into Bruxelles. You'll see arbitrary movements like this.
That most of Europe's current elite are fairly pro-EU, and that the current political bodies are fairly 'on the same page' should not be interpreted as populist favour overall for the EU.
To anyone who thinks that Euroscepticism is a marginal issue, have a look at the Wikipedia entry, it's enlightening, and also gives some insight into all of the different nuances of it.
[1] http://www.pewglobal.org/2017/06/15/post-brexit-europeans-mo...
[2] https://en.wikipedia.org/wiki/Euroscepticism