Centre-back Maguire, brother of Hull defender Harry, and striker Beesley will now become professionals with the League One club.
“It feels very good,” Maguire told the club’s official website.
“When I got the news I was just relieved – now it’s onwards and upwards.”
Beesley added: “I’ve been at the club since I was 14-years-old.
“It’s nice to come all the way through, get my scholarship first and then after those two years I’m delighted now to get my first pro contract.”