I played it on 2006 when it came out, so my memory is quite blurry, but as far as I remember the game doesn't force you to use any specific characters.

I usually used Alicia & Rufus and two Einherjars.

EDIT:
Although I must warn you that some Non-Einherjars characters will leave your party because of the story; you'll get some items depending on their level later in the game.

any character critical to the storyline the game won't allow you to release. everybody else is fair game and won't affect anything related to the story or otherwise.