To decide this, we need to look at the history of the 5th Amendment and how the courts have interpreted it. I'm not a lawyer, but I think it's pretty clear that cyborgs' personal data will be covered.
According to Wikipedia's article on the 5th Amendment, courts have been pretty expansive. You can't even be required to turn over the password to an encrypted hard drive if it would incriminate you.
If I understand the history, the 5th Amendment was partly a backlash over the horribly unfair "Star Chamber" legal proceedings, and also against the use of torture to extract a confession. As a minarchist libertarian, I think it is wise to hold government on a short leash, and I am in favor of keeping the government from taking shortcuts that lead to convictions. But on the other hand, I'm in favor of the truth winning in trials. If you are driving a car and there is a collision, I want experts to be able to examine the "black box" from your car (assuming your car has one); I don't think you can reasonably claim that turning over your "black box" would constitute self-incrimination. So if we imagine a sort of "black box" inside the body of a cyborg, it's hard for me to think that should be private while I think the black box from a car shouldn't be.
Of course, I don't want to see someone have their cyborg body's black box hacked to plant fake evidence against them, but that seems awfully hypothetical at this point.
Hmmm. I wonder if anyone is going to be required to produce the data from their FitBit or other exercise tracker during a criminal investigation anytime soon. I'm guessing that the courts might hold that the 5th Amendment would protect that data. But it would be pretty amazing if you had a guy accused of stabbing someone, and his wrist device had a log showing his hand making stabbing motions at the time the murder occurred!