Quoting from Leung et al, Annual Review of Public Health 18: 83-104, 1997:
Censoring occurs when incomplete information is available about the survival time of some individuals.
So ask yourself: what does survival time mean in this study, and is that information incomplete about some individuals?
If survival time is being defined for an individual as the time between recruitment into the study and the time of an event, then any case (defined in your study as having had an event) clearly has complete information about survival time and should not be considered censored.
It's possible, however, for a study to be interested in some starting time for survival before recruitment into the study. For example, Therneau and Grambsch (page 75) describe a study at a tertiary referral center, for which survival was calculated from the date of original diagnosis. If an individual's diagnosis was made elsewhere before entry into the study at the referral center, then that individual was not at risk for an observable death until entry into the study, and the survival time would be considered left truncated at the time of study entry. If survival time is calculated from some unknown time before study entry, the survival time is left censored. This paper describes some of the issues that arise in these cases.
Whether controls are considered censored depends on their follow-up. Provided that survival time is calculated from study-entry date (putting aside the possibility of left censoring or truncation), if any control individual is followed until the time of an event, then that individual's survival time is by definition not censored. If the event hasn't happened by the time of last follow-up for some individual, then that individual's survival time is right censored at the last follow-up time. So if you do an early analysis of the data most or all controls might have right-censored survival times, but at later analysis times (assuming follow-up continues) some of those might have defined event times.