It's optional, not incorrect, to resay "rules".
Also, I would suggest using "does" to be idiomatic and sound like a native speaker.
I put the optional part in brackets:
A. My country has rules, while yours does not [have rules].
B. My country has rules; your country does not [have rules].
C. My country has rules, while your country does not [have rules].

It's optional, not incorrect, to resay "rules".
Also, I would suggest using "does".
I put the optional parts in brackets:

1. My country has rules, while yours does not [have rules].
2. My country has rules; your country does not [have rules].
3. My country has rules, while your country does not [have rules].

edit: more acceptable sentences:

4. My country has rules, while yours has none.
5. My country has rules; your country has none.
6. My country has rules, while your country has no rules.