Do they have the right to ask? Sure! It's a free country and there is no right to time-off privacy, so they can ask away. A better question is should they ask and do you have to answer?
For the record, I think every time one of my employees took a day off, they said either, "Ugh, I'm getting a colonoscopy next Thursday," or "I'm going to Florida the first week of February!" or something similar. I responded appropriately. ("Ugh is right," or "I'm so jealous. February in Florida is awesome. Are you going to Disney World?") But, if they just said, "I'm taking Tuesday off," I would respond, "I hope it's for something fun."
They then had the opportunity to go into a tale of woe about medical tests, or an evil brother-in-law who did something so terrible that they needed to take time off work to clean up his mess, or they could just say, "Nah, I just have some stuff to get done."
Is there a reason why you can't respond with, "Oh, I just have a few appointments and errands to run"? Is it possible that you are just over-thinking this and worrying unnecessarily? They are probably not really interested in what you are doing on your time off, they probably are just being conversational.
When I asked my employees what they were doing, I wasn't doing it to pry, I was doing it to be kind and because I genuinely cared about them. I would also accept a vague answer as an indication that it was time to stop asking.
[See the 4 big career potholes.]
Now, if your management is taking this to a different level and only approving time off that they agree with, then you've got a problem. If they would allow you a day off for medical testing, but not to go car shopping, then it's a little bit pushy. But, they can still ask and you can just stick with "I have some errands to run."
But, keep in mind that if you are in a company or department where scheduling is an issue, sometimes decisions need to be made. If only one person can take a day off and three people want it, the person who is just going car shopping is not getting it off. So, it makes sense for you to give the real reason.
If you just feel like they are unnecessarily prying you can go over this little dialogue:
The key is to keep emphasizing that you've taken care of any problems that will arise from your absence. A super clueless micro-manager might push and push and then you can respond, "I would rather not discuss what I do on my time off. Thanks!" But, understand that the clueless types are also the easily offended types.
Honestly, though, it's unlikely that anybody really cares that you're having tests. Often, the things that are intensely personal and nerve-wracking to you are no big deal to everyone else.
Suzanne Lucas has nine years of human resources experience, most of which have been in a Fortune 500-company setting. She holds a Professional in Human Resources certificate from the Society for Human Resource Management. She blogs at Evil HR Lady.