Cilliers, 29, won six Springboks caps in 2012 and joins from Montpellier.
McGuigan, 22, has played 59 first-team games for the Falcons having graduated from their academy, while 25-year-old Brady leaves Sale after six seasons.
Meanwhile, Jono Kitto, 23, George Catchpole, 23, and Jack Roberts, 24, have signed new deals at Welford Road.
The contract extensions for the Tigers trio sees them remain with the Premiership side beyond this season, but the length of all six deals have not been disclosed by the club.
Former Sharks and Stormers forward Cilliers is the third South African to join Leicester recently, with former Springboks captain Jean de Villiers making the move in December, while wing JP Pietersen arrives next season.
"At 29, he is coming into his prime as a prop and we look forward to seeing him getting down to work here in preparation for next season alongside guys like Marcos Ayerza, Dan Cole, Tom Youngs, Logovi'i Mulipola and Fraser Balmain," said Tigers director of rugby Richard Cockerill.
Cockerill is also pleased to add "dynamic" talent McGuigan to the front row, but the hooker's departure from Kingston Park was criticised by Falcons boss Dean Richards.
"Over the last few years we have invested a great deal into George's development and had high hopes for his future progression at the club and his development into an international player," said Richards.
"So it is disappointing to hear that he no longer wishes to be part of what we are building at Newcastle Falcons."
Leicester Tigers have signed Sale winger Tom Brady, Newcastle hooker George McGuigan and South Africa prop Pat Cilliers for next season.