Yes! You want native English speaker, not English native speaker.
Here is the reason:
A person who speaks English is an English speaker.
The person (the English speaker) is speaking English in a native manner.
Native modifies "English" and not the person (speaker). A native person is someone who is living in the place where they were born, or the original inhabitants of a country (native Americans were called Indians)
Thus, the correct sequence is native (which modifies) English (which modifies) speaker.