Hello,
I restored my iPhone 4 to 6.1.3, no jailbreak, and the home button is not working, but it is working to reboot or enter DFU!
I tried all tricks found by googling, spending hours, but to no avail, home button just does not make iOS react !
I found out what was wrong! The button contact is shorted, and it is like always pushed in. The button mechanically pops in and out properly, but iOS sees it as pressed, which explains all the strange symptons it had! For now I'll use the white dot that can be activated from the settings, and I'll eventually replace the button contact switch.