Can I install new construction lights into old ceiling with attic vs remodel lights?
If you can access the space above (or in the case where you are ripping off &amp; replacing the sheetrock from below, even if that's not your case) you should be able to use "new construction" style can lights, yes. 

It may involve hot, filthy work in the attic space, and depending where you want the lights and the configuration of the attic, it may be a tight squeeze (or there may be plenty of room - attic spaces vary, a lot.) Depending on your type of insulation, you might be well advised to wear a coverall suit, gloves and a respirator and/or goggles when working up there, which will make it hotter, so beware of getting overheated (passing out in an attic is not a great idea...)

With advancements in LED technology, surface-mount "can style" lights are also a recent option. Look like a can-light trim ring, but don't require a can above them.