If you are specifically looking at the plastic bracket, it appears to be the bracket that is used with some kinds of J-box blanks. It looks like you have a J-box up there currently. Are you planning on actually trying to cut in recessed light housing or are you planning on using a flush mount Low Profile LED like the one in the link. By the looks of the picture, there are J-boxes in the locations that you are suggesting putting in recessed lights which mean you will either have to cut those out and use a remodel can, or use the LED style lights that look and operate essentially the same as a recessed can but are utilized with a standard junction box.

Here are a couple of fixtures that can be installed over a ceiling j-box. They are probably as close in appearance as you're going to get to a recessed fixture. As stated above, you'd have to remove the j-box and replace it with a recessed light if you wanted a real recessed light. I installed the 6" Halo SMD and it wasn't easy because a portion of the fixture has to be pushed up into the j-box...but it looks okay. I had a similar problem to yours: I replaced a flush-mount ceiling fixture and didn't want to remove the j-box to which the fixture had been attached.

• Halo SMD 4 in. 3000K Warm White Integrated LED Recessed Round Surface Mount Ceiling Light Trim
• Commercial Electric 4 in. J-Box 8-Watt Dimmable White Integrated LED Round Flat Panel Ceiling Flush Mount Light