Cook swooped for the 26-year-old after his Brewers contract had expired.
McGurk told the club’s website: “What attracted me was the size of the club and the large crowds that come to watch the games.
“I’ve played against Pompey a few times and I’ve always been struck by how fantastic the fans have been, despite a difficult few years.
“They stick with their side, even when they’re not winning. That’s always impressed me and my team-mates.
“I’ve also played against Paul Cook’s Chesterfield sides before. I liked the way they approach the game and also how much they wanted to play for him.
“When I sat down to talk to him, I could see how much passion he has and wanted to be a part of that.
“I’m excited for the new season. It’s been a stressful few days getting the deal done, but now I’m looking forward to getting down to work and bonding with my new team-mates.”