Gudas is getting a three-year contract, while Verhaeghe's deal is for two years.
Gudas, 30, had two goals and 13 assists for the Washington Capitals last season while leading the team with 164 hits.
"Radko plays an imposing, defensive game," Zito said in a statement. "He is a hard-working defenseman who plays the game with an edge. Radko brings the type of physicality and tenacity to our group that we want as we add to our roster."
Verhaeghe won a title with the Tampa Bay Lightning last season, playing in 52 regular-season games and posting nine goals and four assists. He had two assists in eight playoff games during the Lightning's title run.
The is Zito's first free-agency season with the Panthers, who are looking to build on a season that ended in the qualifying round in Toronto. The Panthers, big spenders in the offseason last year, acquiring goaltender Sergei Bobrovsky, haven't made the Stanley Cup Playoffs since 2016.