I can only answer one of those questions, as I am not a Mason. In theory, a member of the Scottish Rite of Freemasonry has to believe in a “Supreme Being”, so yes, religious to a degree.
Historically opposed to formal ecclesiastical religion in favour of a deistic concept drawing on Old Testament metaphors of the building of Solomon’s Temple to represent the perfection of the member’s character. The French rite admits of atheists.
Massively influential? Once, in certain skilled trades in Scotland, e.g. newspaper printing, shipbuilding, yes, they were strongly Protestant and Masonic affiliations are not possible for Roman Catholics, so it could be used as a sign of group loyalty, but this is going back two generations.
I find it all mildly silly — peculiar clothes, pass handshakes and mock ordeals. Of more interest to an ethnologist (there were other trade societies in the past, e.g. the Horseman’s Word) than to me.
I find the idea of using a self-elected and obscurantist group (ironic, so many early Masons decried the Roman Church as what they slightly resemble now) to rise in society personally unacceptable.
In an attempt to answer your question, I don’t think a Mason would confide in a sweetheart or fiancée about the fraternity, they’re entirely male and not known for being especially progressive.
I think I would make it very clear that you value his candour and honesty with you far above whatever promises he has made to a trade club.
Source: my grandfather on my father’s side was a Mason.