Learn English – What does “I have been sick” imply

Either would be correct in this context.

Generally, the present perfect simple tense describes what has been happened so far, while the present perfect continuous suggests that something is ongoing or unfinished. The distinction isn't strong enough in this question to make a big difference.

Looking at two possible responses to the question might highlight the differences better.

(present perfect simple) I've worked here five years and I've loved every moment of it!

(present perfect continuous) I've been working here five years, but it's only recently that I was transferred to this office.

Similarly, one could say:

(present perfect simple) He's worked three hours today [he might work more, or he might go home after this].

(present perfect continuous) He's been working the whole morning [and he's still not done with his work].

I have been sick for two days.

In practice, this means that you got sick sometime the day before yesterday and are still sick. With the perfect tense the state is still ongoing. However, English speakers are not that specific about the start point of the state, though, so it is possible you are referring to some other day than the day before yesterday.

For most units of time, a good guideline to follow is that "two X ago" can mean anywhere between "one and a half X ago" and "two and a half X ago". For instance, if I said, "I have been living in city x for a year," I could mean ten months, or I could mean sixteen months. If you want to be more specific, you use a more specific unit of time, such as, "I have been sick since Thursday afternoon" or "I moved to this city eleven months ago / in January last year."