One can easily hunt for java deer and wild boar in Mauritius. 

However, one is only allowed to hunt for java deer during the prescribed period, which runs from June to October whereas wild boar or hare hunting is practiced all year round. 

The Domaine du Chasseur and the Domaine d'Anse Jonchée, amongst others, propose this sport.  They also propose other activities and one can stay overnight as well.

But for lovers of this sport, the best bet is to get in touch with a professional hunter who is specialised in this hobby and who has priority access to reserved spots.