I did my own checking on this as well and it seems to me to be false. Here is what I have seen.
1. the twitter account @jojospatafora tweeted that jojo quit, and it claimed to be from jojo's mother. The thing with this twitter account is a few things. First there is only 12 followers which seems a bit weird. Second jojo herself in the superpass interview states her twitter is jojodoll1, which has 77 followers, that tells me that the jojospatafora twitter is fake.
2. It seems the tweets that are posted above have some weird info. It says that Janelle won HOH which doesn't seem possible since CBS says that the returning Houseguests are playing their own game. Now if this was true that Janelle won HOH it would then be that the returning Houseguests are playing the real game and are not there as mentors. It seems more plausible that Janelle won the first comp. which gives her the right to pick her team first.
3. According to the twitter account of Gina as well as Francesca (both look legit to me) JoJo did not quit. Now this can be taken as fact since they seem to be pretty close, or this could be that they are not in the loop on this. If she did in fact quit Big Brother would probably not want the info to get out in order to air it on Thursday, at this point I would think they would have her sequestered until after the show airs on Thursday to make sure it doesn't get out.
In the end I have to doubt it since the rumor seems to have been started by a fake twitter account. I guess the person who made it doesn't look at the interviews because it is asked if they have a twitter account and what it is.