Jennings follows James Franklin and several others to Blacksburg.
Burkle has one year of eligibility remaining.
Shanahan Jr. will follow former offensive line coach Phil Trautwein to Gainesville.
Veidt previously worked with Matt Campbell at Toledo and Iowa State.
Unlike last season, Penn State men's hockey finds themselves in strong position for the postseason at the start of January.
Malloe coached alongside new defensive coordinator D'Anton Lynn at UCLA in 2023.