My aunt and uncle have been British citizens for a few years now. Am I allowed to apply for an EEA extended family visa (with their permission and cooperation of course)? I will be using my Mauritius passport. I am from Botswana and I have a passport from Botswana which I use less due to it being slightly weaker.
This plan is unlikely to work on two counts:
Family of British citizens are not normally eligible for the EEA family permit. It is intended for family of citizens of EEA countries other than the UK. There is an exception, but it requires that you have lived with the British citizen in another EEA country before going to the UK. That doesn't appear to be the case here.
The definition of family member does not include aunts and uncles nor nephews and nieces. Again, there is an exception for "extended family members," which requires that you show you are dependent on your aunt and uncle, are a member of their household, or rely on their care because of a serious health condition.
If you can qualify for both exceptions, then you can get an EEA family permit, but without having more information, it seems unlikely that you could qualify for either.
(There is of course another consideration, which is that the UK is in the throes of trying to leave the EU. If the UK leaves the EU without a deal, which could happen in as little as two weeks, the immigration regulations under which EEA family permits are issued will be repealed. On the other hand, the UK's departure could be delayed by a year or two, or it could remain in the EU, or any of several other things could happen extending EU free-movement law in the UK for some time, or even indefinitely.)