The tourists then got through to lunch on the second day without losing a wicket.
At the interval, New Zealand were 44 without loss in reply to England's first innings 389 -- a deficit of 345 runs.
But they had a moment of good fortune when Martin Guptill, on 24, edged England debutant Mark Wood to first slip only for what would have been the Durham paceman's first Test wicket to be rubbed out when an umpire review to check for a no-ball showed he'd overstepped the crease by a fraction.
England started the second day on 354 for seven, having been rescued from the dire position of 30 for four on Thursday by a fifth-wicket stand of 161 between Joe Root (98) and Ben Stokes (92).
But they lost their final three wickets for just 37 runs in 10.5 overs on the second day of this two-Test series.
Henry ended the innings with just his fifth ball on Friday, an excellent one-handed caught and bowled dismissal getting rid of last man James Anderson.
Fast bowler Henry took four for 93 in 24.5 overs.
England's Moeen Ali -- 49 not out overnight -- pulled Tim Southee for four in Friday's first over to complete a 97-ball fifty including nine boundaries.
But he fell for 58 when he flashed outside off-stump against Boult and was held by Latham, one of three catches in the innings for the substitute keeper.
