It’s unethical to extract a quotation without context, so here we clarify and expand on the quotations used the most, these three;
“We are with Europe, but not of it…”
“If Britain must choose between Europe and the open sea, she must always choose the open sea…”
“We must build a kind of United States of Europe…”
“We are with Europe, but not of it.” (15 February 1930) 
This was pre-WWII, pre-Depression (economic, not his personal one). Churchill was commenting in the American journal The Saturday Evening Post that a “European Union” was possible between continental states but without Britain’s involvement:
“We see nothing but good and hope in a richer, freer, more contented European commonality. But we have our own dream and our own task. We are with Europe, but not of it. We are linked but not compromised. We are interested and associated but not absorbed. And should European statesmen address us in the words that were used of old, ‘Shall I speak for thee to the King or the Captain of the Host?’, we should reply with the Shunamite woman ‘Nay sir, for we dwell among our own people'”.”This is valid to use in context, but clearly his viewpoint altered later, after the Economic Depression of the 1930s, and after World War II.
“If Britain must choose between Europe and the open sea, she must always choose the open sea” (May, 1944) 
Reportedly shouted at the French leader, General Charles de Gaulle, in a raging row on the eve of the Normandy landings in 1944.
“If Britain must choose between Europe and the open sea, she must always choose the open sea. Every time I have to decide between you and Roosevelt, I will always choose Roosevelt.”Churchill had a stormy relationship with de Gaulle and wanted to show loyalty to the US President Roosevelt. Later, it’s said that they made up over dinner and fine wine. It is also highly likely that Churchill had been at the whiskey. Frankly it’s an understandable outburst where De Gaulle is concerned, he was enough to turn anybody against France, or even Europe as a whole.
To take any meaning from it is false.
“We must build a kind of United States of Europe.” (19 September 1946) 
The term ‘United States of Europe‘ was used by Winston Churchill in his speech delivered at the University of Zurich, Switzerland. In this speech, given after the end of the Second World War, Churchill concluded that:
“Indeed, but for the fact that the great Republic across the Atlantic Ocean has at length realised that the ruin or enslavement of Europe would involve their own fate as well, and has stretched out hands of succour and guidance, the Dark Ages would have returned in all their cruelty and squalor. They may still return.No mention is made here of Britain being ‘apart’ or ‘separate’, a point he took care to stress in 1930. So did his view change?
“Yet all the while there is a remedy which, if it were generally and spontaneously adopted, would as if by a miracle transform the whole scene, and would in a few years make all Europe, or the greater part of it, as free and as happy as Switzerland is today.
“What is this sovereign remedy? It is to re-create the European Family, or as much of it as we can, and provide it with a structure under which it can dwell in peace, in safety and in freedom.
“We must build a kind of United States of Europe. In this way only will hundreds of millions of toilers be able to regain the simple joys and hopes which make life worth living.”
Perhaps. Peoples’ views do change. One cannot take any lesson from any one quotation from a man who lived so long and fruitful a life. If any, one must refer to his very last thoughts published on the subject of Europe, in 1957:
“We genuinely wish to join a European free trade area – and if our continental friends wish to reach agreement, I am quite sure a way can be found and that reasonable adjustments can be made to meet the essential interests of all.” Read into that as you see fit.
What is clear is that using any quotation to claim the support of a man who died decades ago is fruitless and unethical.
 Article by Churchill in the Saturday Evening Post, 15 February 1930, quoted in Richard Coudenhove-Kalergi, An Idea Conquers the World, London 1953, pp. 162-163.
 From D-Day: The Battle for Normandy’ by Antony Beevor.
 Winston S. Churchill, Zurich Speech, in: Documents on the History of European Integration, Vol. 3: The Struggle for European Union by Political Parties and Pressure Groups in Western European Countries 1945-1950, ed. by Walter Lipgens and Wilfried Loth, Berlin/ New York 1988, pp. 662-666.
 Address to the ‘United Europe’ movement at Westminster Central Hall on 9 July 1957.