The 20-year-old, who spent time on loan at Grimsby Town during the 2016-17 campaign, was a part of the England squad that won the Under-20 World Cup earlier this summer.
Henderson has revealed that he had the chance to join a number of League Two clubs, but was always confident that he had the credentials to perform at a higher level.
"I'm delighted to come here. I could have gone to a lot of League Two clubs this year, but I had to show that development and go into the next league," Henderson told Shrewsbury's official website.
"I will be working hard every day in training and hopefully performing well in the games before the season starts. I've left no stones unturned, so I'm ready."
Henderson, who joined the Man United academy at the age of 14 after spending six years with Carlisle United, was on the bench for his Premier League club's FA Cup clash with Blackburn Rovers last season.