I am born in the U.K. if I bring my birth certificate is that good evidence that I’m from the uk. Either by Eurostar or flight
In short: No.
Birth in the UK does not automatically confer British citizenship (that's dependent on various other circumstances, depending on your date of birth). As such, a UK birth certificate is not proof of UK citizenship. I don't recommend attempting to use it to enter the UK.
Some comments have suggested you needn't worry about proving British citizenship as Mauritian citizens can enter visa-free. I do not advise this route - that visa-free entry is for specific purposes (tourism, some types of short-term business/educational visits) which do not appear to apply to you.
Acquiring a British passport will make this trip much easier.