Sentence one:
Has any of your employees been sick this week?
Sentence two:
Have any of your employees been sick this week?
They are both correct. Traditional prescriptivism may tell you that "any" is used with the singular because it is thought to stand for "any one." But you will often find it used with the plural. See https://books.google.com/ngrams/graph?content=Has+any+of%2CHave+any+of&year_start=1800&year_end=2019&corpus=en-2019&smoothing=3 -- not all instances of "have any of" are analogous to this construction, but many of them are.
In any event, the subject of the verb has or have is any, not employees. The other answer suggests that the singular is only available if you recast the sentence to make "any employee" the subject, but this is unnecessary. The sentence is not incorrect, but it's not the only way to ask the question with "has" instead of "have."
Neither. The two possible correct sentences are:
("Employees" plural, "have" plural)
("Employee" singular, "has" singular)