Greetings, Hiram1
It's odd that the camera is taking pictures when it's in the process of recording video. If this occurs randomly, even if there's no interaction on the camera by the user then let's try clearing the camera settings to restore them back to factory default to see if it helps resolve the problem. I will provide you with a link to the user guide and the steps to clear the settings are on page 51. Please make sure to have the camera in either the P, Tv, Av, or the M mode to allow this option to be available in the camera’s menu.
https://gdlp01.c-wss.com/gds/9/0300004019/03/eos60d-im3-c-en.pdf
In case the issue persists, then we would have suggested to send the EOS 60D to our repair facility but I'm sorry to say that the facility no longer repairs this model. An alternative option is to participate in the Canon Upgrade Program.
The Canon Upgrade Program option allows you to replace your current camera with a new or refurbished Canon camera at a discounted price. If you would like to take part in the Canon Upgrade Program, please call our Sales Department at 1-866-443-8002 from Monday - Saturday between 9:00 am to 9:00 pm ET. Be sure to have your serial number for your camera available at the time of your call.
