Boxberger, who was an All-Star in 2015 when he led the American League with 41 saves, was a candidate to be non-tendered Friday at the 5 p.m. ET deadline to offer contracts to arbitration-eligible players.
Boxberger, 29, has had each of his past two seasons cut short by injuries, but agent Scott Boras said earlier this month that his client is fully healthy now.
Last season, he dealt with a right flexor strain and pitched 29 1/3 innings in 30 appearances, going 4-4 with a 3.38 ERA. He struck out 40 while walking 11. He is 17-20 with a 3.19 ERA and 44 saves in 231 career innings.
The right-handed Taylor, 22, was 3-4 with a 3.32 ERA in 13 starts for Class A Kane County in the Midwest League. He stuck out 68 and walked 23 in 62 1/3 innings pitched before a shoulder impingement ended his season. He is expected to be ready for the 2018 season.