Henry and Smith suffered calf and side-strain injuries, respectively, during the drawn first Test in Christchurch, which contributed to New Zealand’s inability to bowl out the West Indies in the fourth innings. Santner did not feature in the match.
With the second Test set to begin on Wednesday in Wellington, coach Rob Walter is expected to bring in at least one debutant.
Canterbury seamer Michael Rae and Northern Districts quick Kristian Clarke have been added to the squad.
Clarke, who was part of New Zealand’s recent 3-0 ODI series win over England, has yet to make his Test debut. Rae is being included in a Black Caps squad for the first time.
New Zealand's other seam options are Jacob Duffy and Zak Foulkes, who played the first Test in Christchurch, and Blair Tickner.
Walter will be able to welcome Daryl Mitchell and Glenn Phillips back into the team after they sat out in Christchurch.