I went to private clinic for check ups on both of my pregnancies, and delivered both of them at government hospital. It is not a must to do your check up at gov clinic if you plan on giving birth at gov hospital. Just make sure you have all the records in hand from your visits at private clinics.

For my first kid, I went to private clinics for all the injections. And I planned to do the same for my second kid. It is more convenient since I can go at night without interrupting my daily work. But I cannot do so since there are certain dosage of immunization are no longer available to private clinics (at least to those that I paid visit to). So i ended up going to gov clinics for injections. Best check with clinics if they have those injections needed beforehand to be safe.

Injection at private clinic less than RM100. Sorry i cannot recall exact figure.
Gov clinic = RM2 per visit