Schneider's side are only outside the relegation play-off spot on goal difference having given up a 2-1 lead in a 2-2 draw at home to bottom side Eintracht Braunschweig.
The draw ended an eight-game losing league run for Stuttgart as they collected their first point of 2014.
Schneider was guaranteed his future for the Braunschweig clash but not beyond by the club's board of management last week, and the 41-year-old said he would not quit.
"I'm not somebody who runs away," the former Stuttgart defender said.
"I grew up in the club and can understand the reaction of the fans. I too wear the chest ring (red band across Stuttgart's shirts)."
Schneider understood the fans' frustrations and said they had the right to express their 'displeasure'.
They watched their team go behind on 24 minutes at the Mercedes-Benz Arena when Jan Hochscheidt was left unmarked in the area and side-footed an effort into the top corner.
Alexandru Maxim equalised with a tap in on the half-hour mark and, five minutes later, Stuttgart took the lead as Martin Harnik delightfully turned two men just outside the area before half-volleying into the bottom corner.
The hosts had a chance to double their lead seven minutes after the break but Christian Gentner's penalty was well saved by Daniel Davari.
Schneider felt the penalty would have sealed three points, with the visitors equalising on 82 minutes when Ermin Bicakcic scrambled in.
"You felt the uncertainty of the team at the beginning," he said.
"We fought back well, but unfortunately, in the penalty, missed the chance to decide the game. It's symptomatic of the phase we're in right now."
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