I'd say that 2 days of medical appointments in your first month really shouldn't be a huge big deal. 2 days in your first WEEK would be.
A couple of approaches:
1 - If this pain is really bad and you need to get help from two back to back appointments in one week ASAP, consider adjusting your start date so you can fit in both appointments before you start on the new job. If they ask, you easily say you wanted to take care of a few personal matters in a way that didn't impact starting off work with a great attendance record.
1.5 - if this can't be accomplished logistically, but the pain is still really bad - give as much advance notice as possible - even if that means calling up your recruiter or the hiring manager BEFORE you get into the office, to clear the specific days you need off. It goes much better to say 3 weeks before your starting week "I need these 2 days in my first week off..." than to show up on your first day of work and say the same thing.
2 - Space them out just a little - either do 2 on the same day (often impossible), or do them on different weeks - and if possible, try to book at least one so you can work a half day. There's a big difference between being out 2 days in 1 week and having a week or more of time in between them.
And as @darkcygnus says - less is better when you are new and establishing report (or, many times, even after that). As your old boss said - there can be times when it's hard for a given boss not to be biased, or to cross the boundary between professional or personal - so it's often easiest not to open the door. Often "Doctor's/medical appointments" is really good enough. 2 days really doesn't warrant more.
Now... if you go to the 2 appointments and then find out that you will need a long and time-intensive course of treatment - you may need to then sit down with your boss and set expectations. You still aren't obligated (in most countries) to disclose your health condition, and many bosses won't want you to do so. It's often better to go with the details that most impact your ability to do your job - for example:
I have a minor health thing going on, I'm OK to work at our normal pace, but I need to
get some treatment taken care of. The impact is
that 2 times a week, I need to take a long lunch to cover a 2 hour
appointment - I'm thinking of doing that on Tuesday and Thursday and
then working a bit later to cover the time. I expect that I'll only
need to do this for the next 8 weeks? At that point, I should be able to let you know
if I need to request anything else.
When I had lower back spasms and got physical therapy for them. But it could easily have been all sorts of other things.
With a boss I'm personally close to, I may be asked and/or offer to share more than that - but at that point, personal details should be mutually consensual (ie, only share if both you AND your boss seem to desire it). That sort of choice usually makes more sense when the two of you have gotten to know one another.