Child birth to expat father and mother Indian

Since neither of you are a Mauritian citizen,  I don't think that the child will be granted citizenship if we go by this:

A person is a citizen of Mauritius if he is born in Mauritius after 1st October 1995 and if either of his parents is a citizen of Mauritius.

Anyway, since both of you are Indian citizens, check with your High Commission in Mauritius to know what needs to be done for your child after birth.

As far as education, it's up to you for either public or private school since you work and live on the island.

Like External rightly mentioned, your child will not bear Mauritian citizenship since neither of you are a citizen of Mauritius.
This was only possible for those born before 1st of October 1995.

I don't know about the technicalities of the Indian constitution and what it says about children born abroad, but I would say the most sensible approach to this is to register your child as an Indian citizen at the Indian High Commission.

Regarding education, it's really up to you, you can enroll him/her at either a public or private school depending on how long you will be in Mauritius.