Which is correct What your country name or name?

Neither of those options is grammatically correct. Here's why and the correct way to ask the question:

Why the options are incorrect:

* "What your country name" is missing a subject (who is the "your" referring to?).

* "or name?" is incomplete. It doesn't make sense on its own.

The correct way to ask the question:

* "What is your country's name?" This is the most natural and grammatically correct way to ask.

You could also ask:

* "What country are you from?"

* "Where are you from?"

These alternatives are more conversational and less formal.