(d) More people report themselves as sick than actually are
Nope. The passage makes no claims about who is "actually sick" and who isn't. The passage's only talking about tendencies in how people describe their status in not having paid work.
Consider this: maybe they're all sick, and in denial! Note--in particular--that it puts 'officially' in quotes.
...there is evidence that the unemployed were more likely to report themselves as retired or permanently sick when unemployment rates were high, as both an economic and psychological coping strategy...
It's (perhaps?) easy to imagine people who are filling out forms not wanting to tick the "unemployed" box, and instead calling themselves "retired". Possibly a little less easy to imagine why someone would want to self-label as "permanently sick" vs. "unemployed". But perhaps those people want to plea that they aren't working because they "can't do it" vs. that they are "lazy".
Either way, the people don't have jobs. Hence:
(a) Not everyone who is unemployed reports themselves as such
That not everyone uses the word "unemployed" to describe their status of not-having-a-job is not only suggested, but it is used to support the next statement:
These factors may have resulted in a healthier profile for the 'officially' unemployed in such areas...
That means we must assume there was other profile data being gathered besides the "fake unemployed or retired or too-sick-to-work" categorization. Because if there was not additional data, then speaking about the health profile of the "officially unemployed" would not make sense. With no other data, as all actually unemployed people didn't say they were too sick to work...all actually unemployed people would be "healthy".
This health profile data (that exists outside the paragraph) is the only indicator we have of "actual sickness". The deeper suggestion made is:
"If you're only considering the auxiliary health data of the people reporting themselves as unemployed, that data may show more positive trends in health than truly reflects the surveyed population that is jobless. That's because some are taking themselves out of consideration by not labeling themselves 'unemployed'. Were they instead categorized as 'unemployed', the statistics would suggest the unemployed have lower "actual health" than when they are not included.
If anything, that would suggest that (d) is false! The "actually sick" report their status accurately and exclude their health data from the pool; leading to inflated health numbers for those reporting 'unemployed' status. But who knows; it could be retired people whose additional health profile data is bad and the self-reported sick people are watching SpongeBob and reporting having a great time.