
What actually happened was that The Sun ran a provocative piece on 9 March 2016, claiming that the Queen backed Brexit. The assertion was that she had “rip at the then Deputy PM [Nick Clegg] during a lunch at Windsor Castle”. Clegg rapidly tweeted “As I told the journalist this is nonsense. I’ve no recollection of this happening & its not the sort of thing I would forget” and the BBC ran a story about Buckingham Palace being clear that she was neutral over Brexit.
As the lunch in question was in 2011 it is stretching things to construe whatever was actually said as a comment on the forthcoming referendum. At this distance it might also be quite hard to be sure of what was said and what people wish she had said.
I don’t pretend to know the Queen’s private views on this, but the whole episode raises someone in the role of the Queen might respond to the EU, in terms of the role itself, history and the constitution.

It sounds very familiar: in the UK the proponents of Brexit are pushing an anti-immigration case to “take back control of our borders” while Liberal Democrats are tending to point out the value of immigrants. In narrowly-financial terms, the awkward reality is that immigrants to the UK contribute substantially more in taxes than they take out of the system in benefits. Even the argument to restrict benefits to new arrivals is questionable: if someone comes to the UK, claims benefits while they settle, starts earning and starts paying tax and more-than pays back what they received, then the “benefit” payments look like a prudent investment. The idea’s been pithily summed up in the idea of Schrödinger’s immigrant: simultaneously stealing our jobs and too lazy to work.