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!

I'll just give her the polite smile and nod. When she does not get a response, she will either (1) continue nagging to her heart's content but it'll be to a wall, which is more to allow her to vent her frustration. You can get your peace, quiet and happiness when she has left your place or (2) keep quiet since she's not getting a response.