Nah, they’re nutters with very weird ideas about Britishness, Irishness & everything else, but they no more look to the English for ideas about their own identity & how to manifest it than the English look to NI loyalists as a guide.
If anything they tend to rather despise Englishness, seeing their own culture as the one true, loyal holdout to the Union.
The aprons do seek to emulate Masonic regalia, but the ideals of the Orange Order are entirely contrary to those of Freemasonry, which in any case is not specifically English or even British.