More natural would be to start with "No..." You are not really negating the previous, "thank you" or "nice to meet you", you are just claiming more gratitude or enjoyment. E.g:
"Nice to meet you."
No, nice to meet YOU."
Or:
"Thank you."
"No, thank YOU."
Equivalente a:
"Gracias."
"No, gracias a TI. El placer es todo mio."