McBriar, a 33-year old native of Australia, played eight seasons with the Dallas Cowboys from 2004-11. He played with the Eagles in the preseason and was among the final cuts.
Henry was in his second season with the Eagles. He averaged 48.3 yards with a net of 38 yards on 16 punts this season.
McBriar's average of 45.3 yards per punt ranks fifth in NFL history. McBriar's career net average is 38.4 yards and he has 175 punts inside the opponents' 20-yard line.
Philadelphia also signed long snapper Kyle Nelson to the practice squad and released wide receiver Jeremy Ebert.