I had exactly the same problem on both iPhone and iPad.
A cycle of the apple icon, a brief appearance of a progress bar, followed by the rotating spokes. About half a minute later the cycle repeats. Unresponsive to most button pushing attempts.
I'd tried everything:
1. Bell Canada - "We don;t service iPhones, we only sell them"
2. Future Shop - "We're not authorized to repair, we are only an authorized seller", and
3. 1-800-My APPLE - still not picking up 4 days later and 120 minutes of on hold
4. Of course there are no Apple stores in all of Saskatchewan.
The solution I'm calling - Dumb Blind Luck - worked for both.
While connected to a mac running the latest itunes and OS, press the on/off at the same time asthe home button. Hold them down for a really long time (almost a full minute - appears to power off, then the apple icon appears, and finally the connection icon appears) and this time it actually connects, can be seen by itunes and the restore button can be used.
I love my iPhone and iPad. The thought of being without them was not impressing me.
Good luck everyone
Chris