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!

Like what the above say, i will turned off the ear and knock my head. After all, she don't stay with you too. Whatever she say, on spot you quickly anyhow touch up a bit and knock head and walk away. Maybe pretend to feed baby milk or whatever and went to room, hide awhile also good, hide a few minutes of nag away too.