How is extra health care needed by recruits handled in basic training? For example, say a recruit needs extra visits to the dentist because of poor oral hygiene. How is this handled? Does s/he go in the middle of the day and miss whatever training the rest of the company is doing, or is there time set aside for recruits who need medical care to go without missing anything?
I presume it works like any other dentist or doctor or whatever. You get an appointment, you leave whatever you’re doing to go to the dentist, doctor, etc. These professionals probably book appointments in such a way as to fill their workday just like non-army dentists or doctors.
My brother once talked about the recruits with him in the Canadian army - back then, so many had unaddressed dental problems that a number of the guys in his class of recruits spent a good part of their basic training with quite a few of their teeth pulled and missing, waiting for the swelling to go down and then to get (partial?) dentures. Not sure the advisability of being in a combat zone with detachable teeth that might come loose and choke you…
He misses the day’s training. If he misses too much, though, he will have to recycle to the next training class. For routine care though, that almost never happens. There is usually sufficient time to make up that training before graduation. If the care needed is due to an injury, he will likely have to recycle.
I went through Navy boot camp in 1988, during the 6th week (I think) we were split up to work all over the base doing grunt work. A bunch to the chow hall, the armory, landscaping etc.
The guys that needed something big like wisdom teeth removed had it scheduled during this week to give them recovery time and not miss any training.
Yes, same thing in the late 90s when I was in. I had to have a superficial biopsy, and that was when it was scheduled. Still had to go to choir practice and work on the company flag, though. 
When I was in (did basic in '06) the guys that needed major but non-urgent dental work just waited it out. They were told it’d be taken care of when they got to their permanent duty station. Sure enough, when I got to my own permanent duty station I saw a few guys arrive and disappear for a few days after a couple weeks as they recovered from their own major dental work.