I haven't heard this particular rumor, not like they have In-Q-Tel connections a la Wickr, but they do take funding from the Open Technology Fund [0] which some of the more righteously paranoid take issue with, being US State Department and all. Same parent as Radio Free Asia, Radio Free Europe, Voice of America [1].

[0] https://briarproject.org/news/2018-1.0-released-new-funding/

[1] https://en.m.wikipedia.org/wiki/U.S._Agency_for_Global_Media