My MIL is a very finicky person and she likes to keep things in order in her own home. But as soon as she comes over to our place, no matter how clean it is, she manages to find fault! How do I tell her, she needs to STOP, without sounding rude or crazy!

Tell your husband what is bothering you, and let him handle this and convey it to her mother in his own way that how you get hurt when she pinpoints everything not being okay in the house. Or whichever way he thinks it is fine to handle the situation. I suggest, MILs take everything RUDE if it is spoken by DIL.