I don't know, I'm going to say I don't think that she was induced and the big rush to the hospital with J$ was real. I think if there was a plan to be induced, whoever the mystery person who spent the night with J$, would have already been there when they left. Someone spent the night with him for sure. So they had that part planned, they just didn't have someone to call in a last minute case, which is stupid and irresponsible on their part.
I think when we hear Jenn saying "induced" in the background, she's telling some poor nurse that J$ was induced so she didn't experience all the pain, pain, pain, pain that she did with $liver. Or telling some sort of birth of J$ story. I think if she was induced, they would have been prepared with the camera and recorded it all, because $$$$$.
I think they've lied about a lot of things with this birth, but I think they are both too dumb to lie a big whopper lie like we rushed to the hospital at 6am and he was born 8:31 vs we planned to be at the hospital to be induced. I don't know which hospital she delivered at, I really didn't pay that much attention, but there are many hospitals, very close to their house and at that time of day, it wouldn't take long (not 1 minute though) to get there.