Originally Posted by Dan333SP

You can also spray your water bottle at different points of the shoe or cleat that could be causing the squeak to see if it goes away temporarily while riding. That's how I isolated my strap buckle as the source of my shoe squeak.

Also I'm pretty sure people will think you're putting out so much power that you literally have to cool off your cleats if they see you unclipping and spraying water at the bottom of your shoes.