No, it's absolutely false. If you were not eligible at the time N-400 was originally filed, no continuance can help and there is no way that N-400 can be approved; you'd have to file a new N-400 after you meet the eligibility requirements.
Regarding the situation with the preparer of your N-400. Whomever it was did a very shabby job, but ultimately the responsibility is yours. When you sign N-400, you swear under the penalties of perjury that the information in your application was complete and correct. You should not have signed an incomplete N-400.
Regarding your pending N-400, I can't see much of a chance that it will be approved. It seems to me that the only chance you have there is to convince the IO that your 310 and 146 day absences did not break your continuous residence. You might get lucky, but based on the experiences of other people in this forum, it seems that the IOs usually require a pretty high standard of proof that the continuous residency was preserved. That is, they'd likely want to see things like mortgage or house/apartment rental, completed federal and state tax returns for that period, credit card and bank statements etc (and, if possible, employment). I don't think that an affidavit from your sister would cut it. But since you have already paid the application fee, you don't lose much by trying...
