The Blues slumped to a 4-0 defeat at Wigan last week after conceding 2 early goals and Dunney believes the team should raise their expectations.
"We have to be stronger as a team. We have to be more demanding of each other and less nice to one another," he told the Manchester Evening News.
"As the man who wears the armband I do not want to be losing 4-0 and putting in performances like the one at the JJB.
"We have a lot of respect for each other as professionals and there is a good spirit in the dressing room but we all have too much pride in ourselves to let something like the Wigan performance happen again. It can't happen again.
"We owe ourselves a performance as well as the manager and the fans," added the Blues defender. "There will be plenty of people watching the Middlesbrough game on TV waiting for us to fold and go under again.
"There is our own personal pride at stake. We owe ourselves a few performances. We have to go out and grind a succession of results. I don't think we have ever beaten Boro in the Premiership so tonight (Monday) would be a great place to start."