The answer to your question depends on a couple of things. The first is the definition of EEA national at regulation 2:
“EEA national” means—
(a) a national of an EEA State who is not also a British citizen; or
(b) a national of an EEA State who is also a British citizen and who prior to acquiring British citizenship exercised a right to reside as such a national, in accordance with regulation 14 or 15, save that a person does not fall within paragraph (b) if the EEA State of which they are a national became a member State after that person acquired British citizenship.
So if you became a British citizen after having lived in the UK as an EEA national, you still count as an EEA national despite being a British citizen. But you say you were born in the UK, so this seems very unlikely in your case.
The second thing is regulation 9, which might apply even if you cannot qualify as an EEA national under regulation 2. Regulation 9 is complex, but it requires you to have lived with your partner in another "EEA state" (which is defined to include Switzerland, as well). This is often known as the "Surinder Singh" route, after the person whose court case established this right.
Many people think that the current conditions laid out in regulation 9 are stricter than the EU directive allows them to be, which means that one might be able to challenge it in court. That would probably cost more than paying for the family immigration route, however, and it is not clear to me whether that route would be available after the UK leaves the EU. I would think it ought to be available to people who had moved to the UK while it was a member of the EU, but I am by no means certain of that.
If you have lived your entire life in the UK, then you seem to be stuck with the family immigration route.