Feeling the urge to give personal information

Dear coach,
I have the tendency to give information that I am not asked to, because I feel I am hiding something otherwise.
Today, a colleague told me that in another team, some people are anti-vaccine. I think she does it on purpose but she keeps long silences which gives me time to think and react. I would like to master the ability of keeping silent better, as I am under the impression she is awaiting for me to give information, and I end up doing it.
So I felt it would be dishonest to say nothing, as it felt like she disagreed. And I said I am not against vaccination but I did not get it myself and do not think it is fair to oblige people to do so instead of convincing.
Well, nothing bad as such but I have the feeling her hidden question was, did I get vaccinated?
And I do not think I need to share that information with her.
So my unintentional model would be like:
C – Colleague shares information on people against vaccination
T – She does not approve, and if I do not say anything it feels like I pretend to agree with her.
F – Uncomfortable, untrue
A – I tell her my vaccination thoughts, although Iā€™d rather not
R – I am angry that I was not able to keep quiet, and feel like I was manipulated.

This happens in other situations too, I am not sure why I have this need to justify, especially when people keep silent.