I guess it depends on what one views as "rushing" him, but I don't think they will. First, whatever their internal expectations are, I think they want to at least pay lip service to potentially being competitive, and sticking with a veteran while there's some theoretical hope in that. (And Eli wasn't that bad yesterday, though obviously no one but Barkley escapes blame for a game like that). Second, Eli's a franchise hero, and no matter how non-competitive they are, they're willing to waste some starts on him rather than benching him early, just for the sake of it.
Assuming the team is pretty bad, I think week 7 might be when Jones takes over. They could be coming off a drubbing at NE, with a long week for a little more prep, and two bad opponents coming up. Pretty good situation to bring a new guy into. If not then, I think they wait until after the bye. Jets may be an easier opponent, but that'd be a bit of a media frenzy for a first game. Unless there's a quick reversal from yesterday, I think it'll be hard to justify waiting much longer than that, though obviously a team doesn't ever really have to justify anything if they don't want to. And if there's one bright spot from yesterday's game, 1 sack was well below what the Giants typically have given up the past few years, so maybe Jones will at least have some protection when his time comes.