Well-travelled right-back Regan, 32, counts Barnsley, Hull, Macclesfield and MK Dons among his former clubs and moves to Gigg Lane on a four-month contract having been released by Notts County at the start of January.
Former Cameroon international Mezague has previous experience of English football from a loan spell at Portsmouth during the 2004/05 season, when he was on Montpellier's books.
The 29-year-old was released by Greek outfit Panetolikos at the end of last season and joins the Shakers on non-contract terms having received international clearance at the end of a successful trial period.
Both players are available for Saturday's League One game against Doncaster.
Meanwhile, Bury have released Lenell John-Lewis to allow the popular forward to join Grimsby.
John-Lewis signed on a free transfer from Lincoln in 2010 and went on to score nine goals in 93 appearances - 70 of those coming as a substitute.