Badenhop, 30, had a 3.47 ERA in 63 relief appearances for Milwaukee in 2013 and finished the season with a 2.05 ERA over his last 21 games, allowing earned runs just twice in that span.
He walked just 12 all season and ranked seventh among qualifying NL relievers in fewest walks per nine innings pitched (1.7).
A 19th-round selection of the Detroit Tigers in the 2005 June Draft, Badenhop has played parts of six seasons in the majors, posting an 18-20 record with a 3.98 ERA in 280 games with the Marlins, Rays and Brewers.
Ortega, 20, spent last season in the Gulf Coast League, where he went 3-3 with a 2.45 ERA in 12 games (one start).
The Red Sox roster is now at 40.