A wolf was always loyal to his pack, and Stark were not different. They might have their differences, but they were a family. And they needed to stick together, everything about that moment. Winter was coming and bringing chaos, death and war. And they needed to be ready.
They knew that the Lannisters were heels, ready to rumble and shatter every one who opposed them, but had no choice. The war was started. And there is something about the war of Thrones: Either you win or you die.
And they did not intend would fight for survival, each together so they could one day revive the days of peace in Winterfell.
(But somehow, Ned suspected that this would not happen)