This is a pretty classic mechanically created electrical intermittent problem, when pressed in this location either a connection that should exist is being broken or a short is occurring. It likely is a minor problem and could be as simple as a loose connector however the cause of this isn't going to be cured from outside the camera body. It should go back to Canon for service and include a clear description of the exact issue (and also stick an adhesive dot to where your thumb pressure causes the problem) so that it won't come back with "no problem found" on the service ticket.
"This seems like an odd one,..."
If you have any third party accessories, of any kind (battery grip, flash, trigger, etc) anything at all remove it and deposit it in the circle file.
We all know that Ernie doesn't like 3rd-party accessories, and he's often right. But a less extreme course of action is to remove the accessory and see if the problem goes away. If it does, then throw the accessory away.