It will help if you'll state where you're living. Since you/re using metric measurements I guess you're probably not in the US.

I live in Canada. Inner height should be not less than 0.6 inch or 1.5 cm.

I don't understand how these junction boxes would be used -- but for LED undercabinet LED lights, have you considered a product like this dals LED undercabinet linear fixture? They come in various lengths (12" example) and there are connectors available to link multiple units together so you'd need only one power source. Sold under the "illume" brand at Home Depot (ca). An alternative is LED tape lights (example) which can be cut to length and linked together.

Thanks DavidR, what would be good examples of "A 4" square box 1-1/2" deep" and where can I buy them. I would really appreciate if you can send me a link. Thank you

I would strongly prefer for this box to look nice and be in white color since it is going to be visible under the counter. Maybe made from plastic too.

Now there I'm afraid I can't help you. I can suggest what might work here in the States, but I'm sure the rules are different where you live.

When I go to Europe, I look at houses there. French people move out and they take the light fixtures with them! The cables hang out of the ceilings, with no boxes, even in recent installations. The wire colors are different.

In an old house I saw an ancient switch hanging from a cable over the head of the bed, at just the right height that you could turn off the room lights after you're snuggled in for the night. Now, that's civilized.

Main panels there look totally different from ours. Often they hang right out in the open, on kitchen and living room walls, with computerized meters beside them. A 60 amp main there is like 200 amps here. They might have 8 branch circuits, maybe 12.

It's an electrical world I don't know. I'm kind of lost there, but I'm a geek, and I'm fascinated, and I want to learn more. I must be the only tourist who's at least as eager to check out the hardware and DIY stores as the museums.

But I digress, sorry.

I think Canada is probably closer to the US in its electrical practices and codes, but I just don't know. I hope that someone else here knows.

I can only suggest that you head for the public library and do some reading. Visit job sites where houses are being built and wired, if they'll let you. Go see the jobbers where the electricians buy their materials, if they'll talk to you. Maybe there are websites aimed at Canadian electrical DIY.

Getting back to a point I made before, maybe your code says that some voltage under X volts (48 volts here) is intrinsically safe, and you can use any junction box you want. Then you can put a power supply somewhere safe and run low voltage wiring to your LED lights. In that case you can probably (check the codes!) use anything reasonable for your junction boxes, such as plastic electronic project boxes.

Sorry I can't be of more help.

Here, plastic boxes are usually blue or grey, and probably can't be painted very well. I would use a metal box for that reason. Of course, if you use low voltage wiring, you don't need a box at all!