The 21-year-old, who departed Coventry City in the summer, had been on trial with the Championship club over the last couple of weeks.
Boss Darren Ferguson told the club's official website: "Shaun has come into the building and performed very well.
"He is a good size and I have made a decision based on his performances in the two games I have watched. It gives us an opportunity to develop the lad.
"We have liked what we have seen, he has settled in well with the players, which is important, and he will get an opportunity to impress us. If he does impress us we will look to extend his contract. It is as simple as that."
Jeffers turned down a move to Tranmere Rovers earlier in the summer.