Most, if not all, conferences have a rule stating that if a certain number of conference members sponsor a sport, the conference must offer a championship in it. I have a feeling that it's six for the Big Ten since there's at least one other Big Ten sport with six schools (Field Hockey).
I heard about this as well from the Big 10 and 6 sports but never did the research to confirm it. It will be intresting to see if any other Big10 school upgrades or if ND joins and then what would happen.
Conversely, the NCAA also has a rule that you must play a sport in your conference if the conference sponsors it , unless if you're playing the sport in a different division or subdivision (not sure how Notre Dame football avoids this).
Are you sure about the MUST part? I thought the rule was that you cannot participate in another LEAGUE if your conference sponsors that sport. Since ND is independent in FB, it doesn't affect them. This would be an issue if say ECU or UCF gets a FB only invite to the BE because they would have to move their sports to a league that doesn't offer Bowl Championship Subdivision FB (Atlantic Sun or Southern Conference, for example).
As you note above, this wouldn't be an issue with Army or Navy because they have their sports in the Patriot League which doesn't offer the 'Bowl Championship Subdivion' FB.