S.G.
My son is a private pooper too. He does not like anyone in the bathroom with him when he goes. He even tries to kick me out of public bathroom stales!!
I know this has nothing to do with your problem of the rug, but you need to make your son understand that this is not ok and next time there will be a punishment for it. He is well old enough to know what it feels like to need to go poop and to inform and adult so he can be taken to a bathroom. This is a power struggle for your son and he is winning. It gives him a form of control over something that you want him to do.
Now on to the rug, even with the rubber backing (I'm assuming like a bathroom rug) you can put it in the washing machine with some laundry soap as a pre treat or shout. A front load machine would be best but a top loader will do as long as it will fit. If it's a big rug, take it to the laundry mat and put it in one of those super jumbo machines. Wash on cold and let it air dry. Putting it in the dryer is what will hurt the backing.
Hope this helps and good luck with your son!
S.