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I just tried putting on the molded mud flaps for my wires 2013 explorer. The rear's went on NO problem.
Jacked up the front and removed the front wheels and the 2 plastic screws that you need to remove and re-use won't come out. It is like they are stripped. I have tired applying some pressure with a screwdriver to get the threads to engage but nothings seems to work. I don't want to break them.
Try putting the blade of a pocketknife under the flange of the screw and pry it gently as you turn it with a screwdriver. If the screw has threads, this will help them engage. It will work best with an impact driver, but being a plastic screw, you'd have to be careful.
It sounds like those are not screws at all. I agree to tru and get under the edge od the "screw" and pull out. They sound like the push-in typ pins that spread the fastener below. good luck, Jag