Are you sure this guy is not just an idiot and hit some blocking, duct or something. There is no way that the ceiling/floor is so shallow that it won't allow a junction box to be installed. It would not be able to support any weight at all. Are the structural members exposed (joists, beams etc) and he is drilling between them into the subfloor above? It doesn't add up.

The cables used for permanent wiring use clamps were they attach to a box. Some fixtures are also designed to have a clamp installed so that a cable can enter directly. The hole would be around 3/4 inch, usually at one end of the fixture. The problem with smaller boxes is box fill. The NEC requires a certain amount of volume based on the number of wires, devices, and internal box clamps inside the box. The thinnest box for general work is about 1.25 inches deep and 4 inches square. You can install a plaster ring (making the overall depth 1/75 inches). If the box is in a joist cavity there are mounting methods that can recess it deeper unless something structural is in the way.

Our soon to be forever home was built in 1989 also has track lighting in the kitchen. The electrician said that recessed lighting was out due to the cathedral ceiling and not enough clearance. It's also an open plan. I started looking at track lighting that was more decorative and more expensive but decided that, in the end, I didn't want the track lighting to be a focal point and to detract from the other pretty lighting fixtures in the room. So....after that long winded explanation, I decided to get a low profile fixture that fit in the existing track and - voila! - we have more light and the track lighting isn't really noticeable anymore. Bought the fixtures at Lowes for $8 each. The old fixtures looked like this except the inside was black which made them look even uglier and noticeable: These are the new ones and - it's hard to tell from this picture - the profile is about 1/2 that of the old ones. Updated and easy on the budget.

The 0.87” is the depth of the diffuser and how much it protrudes. They are mounted on the underside of the front cabinet lip. The size didn't influence the decision on where to place, the installation was delayed due to the issue with the Armacost strips and finally happened in the middle of some other renovation chaos so honestly, the only thing that influenced the decision is the electrician lol. He just put them in where he usually does I guess and it wasn't until a few days later when things settled down that I thought about placement. It was a fleeting thought anyway because they light up the entire area(s) very well so if they aren't in the "preferred" spot that others would suggest, I have no idea and it doesn't bother me since they work perfectly well right where they are and do exactly what we need them to do. You're welcome! I'm happy to share my experience - there are so many things to think about and consider, and I received help from others and this forum so I'm happy to try and give a little back when I can. ETA: I just looked up the exchange with Inspired LED and they indicated that when using the the strips, they need to be installed facing the backsplash to allow the LEDs to wash and blend together and create a more even beam of light. With the diffusers, that isn't necessary since the panels give that even beam of light, so they should be installed facing directly down toward the countertop.