Hauschka signed a four-year, $12.4 million contract, according to a source. He will replace kicker Dan Carpenter, whom the Bills released earlier this week.
Hyde signed a five-year, $30 million contract with $14 million guaranteed, NFL Network reported.
Hauschka, 31, has a strong leg, but he has struggled recently with extra points, missing 10 PATs over the past two seasons. Seahawks coach Pete Carroll often pointed out that Hauschka, who had been with the Seahawks since 2011, was kicking the ball too low.
Hyde, a former fifth-round pick from Iowa, was one of the Packers' most versatile defensive backs in 2016, playing everywhere from safety to the nickel and dime slot positions. Hyde, who turned 26 in December, even played some at outside cornerback when injuries hit Green Bay hard last season.
The nine-year veteran, however, has been good on kickoffs, and he hit nine of his 11 field goal attempts from 40 yards or longer in 2016.
Hauschka, who broke into the league with the Baltimore Ravens in 2008 and also has kicked for the Denver Broncos, has made 87.2 percent of his career field goal attempts (191-of-219). He earned a ring when the Seahawks won Super Bowl XLVIII.
Hyde finished the season with four interceptions over the final seven games, including the playoffs. He has nine career interceptions.
Hyde, who also has three career punt returns for touchdowns as a core special teams player, has missed only one game in his career. He played all 19 games last season, including playoffs, but left the NFC title game because of a shoulder injury and did not return.
The Bills also officially announced the signing of fullback Patrick DiMarco and announced contract agreements with offensive lineman Vladimir Ducasse and safety Jordan Poyer.
ESPN's Sheil Kapadia and Rob Demovsky contributed to this report.