James Boswell's uncle John was a member of a sect known as the Glasites.
A Mademoiselle or Countess of Boetzelaer is mentioned twice in Boswell in Holland, first on the List of Ladies Given Boswell by the Countess of Nassau (BiH, p. 69) and again (although possibly a relative of the former) in a letter from Archibald Stewart to Boswell of December 30, 1763. Stewart writes (half jokingly) about the fine ladies of The Hague saying to one another [...] "it is evidently that young English lord recently come from Utrecht who Mademoiselle Boetzelaer told us was full of wit, very gay and extremly likable".
The Countess of Boetzelaer (or Mademoiselle Boetzelaer) in question is probably either Magdalena Elizabeth van den Boetzelaer (1730-1800) or Sophie Louise van den Boetzelaer (b. 1731), both daughters of Willem Lodewijk van den Boetzelaer, Vrijheer van den Boetzelaer (1691-1733) and Eva Maria Slicher (1700-1780). Another possibility is, of course, that the Countess is their mother, and that the Mademoiselle in question is one of the daughters.