The 20-year old spent the first half of last season at Grimsby Town, who was managed by the current Shrewsbury boss, Paul Hurst.
Henderson was recalled from the loan spell early after United third-choice keeper Joel Pereira went down with injury.
Speaking to Town's website, Henderson said: "I'm delighted to come here and be reunited with Doigy and the Gaffer.
"I had a few options, but from my point of the view the interest the Gaffer has shown in me was massive.
"For the last six months now he's been onto me saying he wants me here this season and being wanted is a great feeling, so I'm really delighted."
Henderson has yet to make a senior appearance for the Red Devils.