I have a jailbroken iPhone 4 white 16gb

the phone is not charging while on and is not being recognized by iTunes BUT it is charging when its off/ dead for some reason. I took it to the apple store today without an appointment but they tried multiple official cables and none worked so the cable is not the problem.

I never opened up this phone. I want to take it back to apple so that they can give me a new phone but the problem is that the phone is jailbroken using Jailbreakme 3.0. I can't restore because its not connected to iTunes and its not restoring from Settings > General > reset its hanging on the apple screen with the progess bar

So can Apple genius bar tell if the phone is jailbroken if there is no Cydia? but of course the internal packages remain oh and also in settings there is still Cydelete in settings.

If it's hanging on the apple screen with the progress bar you should be fine to bring it in to the genius bar. I've had them replace phones for me that worked that had all of the jailbreak apps just hidden with Poof before, even though the diagnostic tool was spamming them to let them know I was having crashes with Cydia and ScummVM