On my Canon 60D, under the flash control menu, when I select the external flash function setting, I keep getting
this menu cannot be displayed. Incompatible flash or flash's power is turned off. How do I get my camera to recognize this 3rd-party external flash?
Your camera's menu cannot recognize the flash, but it will still work. If you put the YN-560III on the flash hotshoe and push the shutter button, the flash will still fire in sync with your camera shutter. But you have to set/adjust the output power manually on the back of the flash. If it's not firing, check the batteries, and that the flash is seated correctly on the camera hotshoe (yes, it's possible to put the flash on backwards or mis-seated so the pin doesn't hit the sync contact).
The YN-560III is a manual-only flash. If you look at its foot, you'll see it only has a single pin. That pin is for the sync ("Fire!") signal of the flash.
All other communication requires the additional four pins/contacts of the Canon hotshoe. TTL, HSS, 2nd-curtain timing, and communication via the camera menu all happens on those other four pins your flash lacks. All that camera menu message is telling you is that the camera isn't sensing anything on those four contacts. Only eTTL-capable flashes can talk to the camera menu.
But not all eTTL-capable flashes. I own Canon's 580EX and 430EX (Mark I) flashes that were created before the Digic 4 processor and the flash control menu was introduced in Canon cameras. Either of those flashes cannot be controlled via the menu, either: my Canon 580EXII and Yongnuo YN-568EX can.
If you want to control the flash through the camera menu, you have to get a flash that has this feature.