My insurance did not cover genetic testing, but I did not end up paying anything. I don't know if there are other companies doing genetic testing for thyroid cancer than the Afirma/Veracyte, but that is what I used.
Even if you have insurance, if your insurance will not pay for the testing, and your income is under a certain level (I think it was income under 100K a year, but I could be off with that amount, this was 3 years ago for me), you can qualify for the Veracyte Access Program. That's what I used and I ended up not having to pay anything. If I'm recalling correctly, it was very easy to apply for the Veracyte Access Program, but I had to contact them using an email address on the website to apply for it. I think maybe I had to mail in a tax return to verify I qualified.
It is beneficial to insurance companies to cover the genetic testing since if the test comes back benign, they will save a lot of money by not having to pay for an unnecessary thyroid surgery. Hopefully your insurance will cover it. Aetna is listed as an In-Network insurance plan on the Afirma website (but I know how that goes - my in-network hospital for my in-network thyroid surgery somehow had out-of-network pathology department that I then had to get them to approve).
If your insurance doesn't pay and you don't qualify for Veracyte Access Program, it looks like it costs around $4800 now. But, I think it is likely that your insurance will pay for the test.