Bruce is an old fashioned businessman. His father wanted to better himself and his family, and Bruce was sent to stuffy old fashioned private boarding school. These places had barely changed since the heydey of the British Empire, all ritual and hierarchy - stiff upper lip, don't let the side down old chap, and king and country, yessah! There was widespread respect, if not admiration, for hereditary gentry. Well-connected military figures with strong resolve like Churchill were considered ideal role models for boys back then. The archetypal Englishman, if you like. Even though he didn't fit in and was eventually kicked out, he learnt to walk the walk and talk the talk, and it's very likely that some of that mentality rubbed off on him. He certainly plays the role. That background's got very little to do with present-day American ideas of political conservatism, and I don't see any direct link between his opinion of the likes of Churchill and his view on EU membership.
As I said in the Brexit thread, Bruce is one of that increasingly rare sort of businessmen who has an interest in investing in something for modest profit only, and running businesses that do or produce something useful. He seems to have a few old style business principles. Of course businessmen don't like regulations and multiple layers of bureaucracy, because it gets in their way, and they have to bargain with the authorities to be able to get what they want at times. It becomes a game. Businessmen also get frustrated with the way that public sector decisions take ages to come about, because they're so procedure-bound.
Bruce is also huge on marketing. He's trying to sell Britain as a brand in all of this, and be ultra-positive about the future, at a time when it's very clear that the country's leadership is hideously disorganised. He's harking back to past glory days because it's an image of strength, decisiveness and influence.
For the record, the NHS isn't a third class healthcare system that routinely endangers the lives of cancer patients. Basic private healthcare is practically identical to NHS healthcare unless it's for a very niche service. Only gold-plated healthcare packages would be significantly better, and part of the attraction there is exclusivity.