From a purely physiological standpoint, the answer would have to be no. The boys were on the island for less than two weeks and could have survived on the fruit they found since they have fresh water. They likely would have struggled and been less healthy given the lack of protein, but they would have been able to survive.
The other side of the question is whether they really needed Jack to get meat or could they have depended on others and the answer is clearly that other boys would likely have been able to step up and manage the hunting duties.
The one thing that Jack does figure out and take advantage of is the understanding that the boys' fear can be channeled into hunting and used to maintain control. No one else on the island appears to have this understanding and the desire to use that fear and to do things like painting faces and using the hunt (and the pretend hunt) to allay their fears and channel their energy into other things.