I don't think you're overrreacting, but rather than complain as such, I would try to educate. I would a) try and pick a quiet moment in the surgery (ha!) to have a word with the receptionist, nicely. Or write a note to the receptionists if that isn't possible.
Something along the lines of
"Dear Receptionist,
Let me start by saying you usually do a super job, often under pressure and deal with many demanding situations in a day's work. However I would just like to highlight something that you may not have thought of.
Recently I was given a note by the doctor to bring over to reception to book in for a vaginal swab and a smear. This information was written down, however you asked me out loud "So you want to book in for a swab and a smear" within earshot of other people in the surgery.
I did feel uncomfortable at having this information read out aloud, for patient confidentiality reasons and also because the content was of a very private/personal nature.
You also asked about dates when this appointment could be booked. Obviously, there are certain days on the female cycle in which a swab and/or smear cannot successfully be performed, so I said those dates wouldn't work due to my cycle. I felt uncomfortable having to explain out loud that I would be on my period on the days you were suggesting.
I understand that it may not have occurred to you that reading out my note about a swab and a smear would be embarrassing for me/make me feel like my patient confidentiality had been compromised. However, it did. I am sure that you wouldn't want a patient to feel this.
Once again, I appreciate the work that you do in the surgery. I would just like to bring this matter to your attention so that the service can be improved even further.
Yours sincerely,
OP.
*Having written this, I actually feel it's a darn liberty and you should throw the book at her
I'd be fuming. But the right thing to do is try to educate in the first instance, and people are usually more willing to listen if their good work is pointed out as well as their, um, bad work.