John Henry's "brother" tried to kill him and also wants Savannah dead. Those are two good reasons for John Henry to want to stop him. So why jump to the future to destroy the AI that's after him and Savannah in the present? Well, I don't think he did. I think John Henry is still in 2009 and after finding out Kaliba's location from the water cooler terminator's chip, he went after him.
Weaver seemed certain he'd jumped to the future, so I'll assume John Henry fooled her by making it look like he'd jumped. He could've messed with the display screen or maybe even sent an empty time bubble ahead to make it more convincing. She had told John Henry that Savannah's survival depended on his but his survival didn't depend on hers. That implies that Weaver was really only concerned about protecting John Henry and I think we can be sure John Henry would want to try to protect Savannah.
If Weaver knew he'd gone off to confront his brother, she'd probably find him and bring him back. She'd already asked Murch about packing everything up, so John Henry knew she was prepared to move him to keep him safe (A "mother" moving her "son" to keep him safe seems rather familiar, doesn't it?). But he didn't want to run away, didn't want to hide and didn't want to leave Savannah at risk. So even though he wasn't yet prepared to fight and maybe didn't know what kind of situation he'd be heading into, John Henry faked-out Weaver. I believe he walked out of Zeira Corp to go have a "computer vs computer" showdown with his Kaliba brother — who meant him harm and meant Savannah harm.
If it doesn't fit, you must... use a different chip
Cameron's chip would not have fit John Henry's port. She's a different model and uses a different size/shape chip than Cromartie (now John Henry) did. I think we all just sort of rolled our eyes at that and figured we had to accept that he made it work somehow. But maybe he didn't. I figure we've got two other chip possibilities: water cooler terminator's and Vick's. We really have no idea what happened to either one. Water cooler terminator's chip was damaged, so that seems unlikely. I'm not counting it out, though. John Henry might have some mad chip repair skillz! Of course, if his chip skills were that good, he could probably make Cam's chip work for him and I don't like that idea.
Last we saw Vick's chip, it was working fine. I can't imagine John would've given it back to Cameron after he mucked around with it, but she could've pocketed it again at some point. I certainly don't ever recall seeing it destroyed. Cameron sure seemed intrigued with Vick's chip when she first pulled it. Maybe she actually started plotting this back then? Or maybe as far back as the "will you join us?" circa 2027. Either way, she held onto that chip, hid that chip and never gave a valid explanation as to why (if it was just to get info, then why keep it a secret?).
If John Henry didn't jump to the future, then neither did Cameron's chip. And if he didn't even use her chip, it's still somewhere in that room. I'm going to say it's in her pocket (that she told JH to put it there). Remember that super cute greeting between Cameron and John Henry? I'm thinking they had some friendly chit-chat and compared notes... about John, about Savannah, about Sarah, Weaver and Kaliba. And they devised a plot. But why would Cameron want John to jump ahead?
"I'll stop it."
You may recall that I was not so fond of Born to Run. I could not (and still don't) accept Sarah's lack of fight over John jumping away. So I've had to grab onto the notion that she figured he could be free — his fate could be changed — if she let him go and she stopped j-day. He'd land in a normal world and finally be able to live a normal life. In short, he'd be happy... and safe. Since Sarah and Cameron had a common purpose (protect John), I see no reason why Cameron couldn't have come to the same conclusion — that the best way to keep John safe was to send him ahead with the belief that he'd be safe when he arrived.
It seems to me that Cameron's manipulations were intended to get John emotionally attached to her and to convince Sarah that John was better off without her (Sarah). And I think she succeeded on both counts. But her mission is only partially completed. With John out of the way, she and Sarah ("you're the best fighter... the best") can get down to business stopping Kaliba, preventing j-day and ensuring John's safety.
You all put back together, Tin man?
Things would be crazy right after the time bubble left. The whole freaking building may be about to come down on them and police and the fire department are no doubt on their way. Sarah needs to get the hell out of there. That fuckhead Ellison would probably be sensible and try to rush her out. But I'm thinking that after a moment of being stunned, Sarah would come to her senses, grab the Turk and also want to take Cameron's body (can't leave an endo behind to be found). That's when she'll find Cameron's chip. Since it's her only hope for finding out what the hell is going on, she'll put it back in (also Cam walking herself out is a hell of a lot easier than having to be carried).
The metal, the merrier!
John Connor took off to the future with Weaver and John Henry stayed in the present where Sarah is. Allison is with John in the future; Cameron is with John Henry in the past. John is going to hang with the humans and maybe the lessons his mom taught him will start to sink in (people matter, machines don't feel things and don't know love). Meanwhile, Sarah is going to find herself working closely with two machines (Cameron and John Henry) which will maybe help her start to understand her son a little better.
Isn't it interesting that John's new name was going to be Henry?