Boult's new ball partner Tim Southee is already out of the Test in Hamilton with a torn hamstring and the Black Caps are also missing batting stalwart Ross Taylor.
Boult missed the second Test, won by South Africa by eight wickets, because of groin soreness and chief selector Gavin Larsen said he had failed a fitness test.
"Trent did everything possible to get himself ready for this Test and is extremely disappointed to be missing out," Larsen said.
With first-choice backup Lockie Ferguson also injured, the Black Caps called up uncapped Central Districts all-rounder Scott Kuggeleijn as cover.
Larsen said Kuggeleijn, 25, had consistently finished among the top wicket takers in domestic cricket in recent seasons.
Kuggeleijn was unavailable for national selection until recently while he stood trial for rape over an incident dating back to May 2015.
However, a jury found him not guilty last month, deliberating for less than an hour before clearing him.
"Not having Trent or Tim available tests the depth of our squad, but we're confident of the job that Neil and Matt can do for us," he said.
New Zealand trail 1-0 in the three Test series and need a win in the final match starting in Hamilton on Saturday to draw level with the Proteas.
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