Should I have CSI during periods I was unemployed?
If you registered for Job Seekers’ Allowance during those gaps of unemployment, you wouldn’t need to have CSI.
If you registered with the Job Centre in between employments, you will have maintained your worker status.
It is also worth querying whether an EEA or Swiss family member was exercising treaty rights by being a worker at the time of your unemployment so that you could be considered a lawful resident by deriving your rights from them.
Note: this question, like other FAQs on Comprehensive Sickness Insurance (CSI), relates to applications for British citizenship, when considering whether periods of past residence were lawful.