That's a shame you are having problems with the focusing. It's rare, but things do break down and require repair. I hope it's not expensive.
The other suggestions of things to check are good: MFD setting on lenses with that feature; whether or not the AF "Hunt" feature of the camera is enabled, etc.
I don't believe anyone has mentioned it yet.... Also whenever there's a focusing issue it can sometimes be solved by cleaning the lens-to-camera contacts. Just use a few drops of isopropyl to dampen a clean cloth and wipe the contacts with that... both on the rear of the lens and just inside the bayonet mount of the camera. Of course you have to be a little careful to stay clear of the lens optics and the mirror in the camera. But the idea is that sometimes oils get on those contacts and can partially interrupt the small voltages used for communication between the camera and lens.
While you are cleaning, also check the contacts inside the camera. They are springloaded "pins" of a sort, and sometimes get stuck pressed in. If that happens, it can lead to intermittent contacts and effect camera-to-lens communication... also might effect focus (can show up as problems with aperture control). If this is happening, it might be something you can fix yourself, or may need to go in for service.
Honestly, it's unlikely that a "reboot" of the camera would help in any way with a focusing problme, so the fact that the onboard (memory) battery isn't user removeable on 60D probably is a moot point. (Still, not sure why Canon made that change with the 60D. It seems sort of short-sighted of them.) The camera likely isn't old enough to need a replacement of that battery, either. (And even if that little memory battery were dead, all you lose is the date/time settings, and perhaps some of the custom functions or menu settings whenever you turn off power to the camera or remove the main battery.... otherwise you should be able to use the camera only with power from the rechargeable battery.)
Anyway, just a couple more things to check. It would be great if it corrected your problem. Maybe it will, maybe it won't. But at least it's free and easy to do the cleaning and check the springloaded pins.