Bodine went No. 111 overall to Cincinnati, Boston went No. 128 overall to Carolina and Price went No. 225 overall to Minnesota.
Bodine, a native of Scottsville, Va., was a two-year starter at center for the Tar Heels.
As a junior in 2013, he started all 13 games on the offensive line, primarily at center, but also saw significant playing time a guard. He earned All-ACC honorable mention accolades for his efforts and elected to forego his final year of eligibility to enter the draft.
"I want to come in and compete and give everything I can to the team," Bodine told reporters following his selection. "I have high expectations for myself, so obviously it will take a lot of work to transition, and any starting spot will take just as much work to earn that. I'm definitely looking forward to getting in there and giving it my best and seeing what happens."
Bodine led all players with 42 bench press reps of 225 pounds at the 2014 NFL Combine.
Boston, a native of Ft. Myers, Fla., finished his career with 13 interceptions, which ranks third all-time at UNC behind only Dre' Bly (20) and Lou Angelo (16).
"It's a dream to play on this (Panthers) defense with how great they are," Boston said. "Just talking about it I'm just getting chills because I know I can learn so much from those guys."
He earned second-team All-ACC honors at safety after finishing the season with a team-high 94 tackles and five interceptions. He started all 13 games at safety and helped lead the Tar Heels to a victory in the Belk Bowl over Cincinnati.
Price, a native of Pompano Beach, Fla., earned All-ACC honorable mention accolades in 2013 after finishing with 80 tackles, 4.5 tackles for loss and nine pass breakups. The cornerback 6-0, 200-pound corner started 30 games during his career, including all 13 games last fall.
James Hurst and Bryn Renner did not get selected in the draft, but were coveted as undrafted free agents. Hurst signed with Baltimore and Renner with Denver.