I don't understand what's their problem with the B1G? That seems like the perfect spot for them smack dab in the middle of the conference footprint. I mean they're a stone's throw from MSU, Michigan, Purdue, Indiana, Northwestern, Illinois, and even Ohio State. They are in the middle of the country and even with the coast to coast nature of the B1G, travel would be relatively easy for them because they'd never be more than a 3 hour flight from any of the schools. And the B1G pays well. I don't get it.
ND doesn't want to join a football conference, any conference, ever.
It is a long story and is tied to ND's particular history.
ND wants independence for independence sake. ND thinks it is central to the identity of the school, not just the football program.
ND thinks that being a football independent is best for the brand...of both the football team and the school itself.
ND describes it as being part of ND's DNA.
It simply sees no reason to give that up to join a football conference, any football conference.
To directly answer your question, four or five generations of ND fans are taught from birth that the Big Ten tried to smother ND's football program, first by rejecting it, then by trying to organize a boycott of it.
That led to Knute Rockne coming up with the novel idea of barnstorming all over the country as an independent.
Since then, the Big Ten has tried to end ND's independence and "checkmate" it into joining the conference against its will. It is still doing that.
So, there is a long standing, deep abiding hatred and distrust of the Big Ten by ND people.