No no no no no no no. No.
Unless I'm missing the point of Brink, different classes are required to do different but equally useful parts of the mission. Although class specific characters will be powerful, they will also be an immense handicap to your team if you refuse to switch to a different class simply because your character is "class specific".
If, by class specific, you mean that you will focus on developing skills in a particular class more than the others, but will still play all classes with that character, then yes, it should be fine. But because there are only eight people per team, if 5 of those eight are all soldiers and refuse to change their class, it is going to be one heckuva bad game.
See what I'm saying?
That's if you refuse to switch though, it's ideal to have a few players setup for more than one class with the rest specializing to increase the odds of victory. When I play with friends I'll be using a Specialist that works best with the team, when I'm on my own I may have a mixer or I may just have a specialist and do the other objectives if need be.
I think the hope with the game is that people will find a role and we'll get people to fill all needed objective roles without having to switch, being able to switch is SDs way of making sure you won't lose just because you showed up with the wrong set of randoms.