Your Kayfuns will run on pretty much any box mod with a 510 connection, although most likely your button tops won't. Those are usually used in more specialized equipment these days, most of it older. The majority of box mods will require two, flat top 18650 batteries. The good news is, depending on your charger, it should probably be able to charge flat tops (what do you have?) and many flat tops that will probably suit your vaping applications (if you are a MTL vaper and really require more mAh than runtime, those batteries are not insanely expensive, from an approved vendor.
There are several good box mods and what you get will depend on how much power you need, do you want a DNA device that can provide accurate TC, how large or heavy you want your box mod to be. There are tons of great options, I am partial to the Eleaf Invoke, it's one of the smallest, lightest two battery mods around and doesn't sacrifice build quality in the process. Teslacigs is a company that is very reliable (based out of Britain, I believe) and they have a very efficient, smooth vape despite not having a DNA option (I don't think). Another two battery mod I love is the Smoant Cylon, great vape, lots of great features, and great handfeel. I'm also partial to Innoken generally although for the most part, I have used their tanks, but quite reliable. The options can be dizzying you may even wish to just check out some of the VW/VW and TC forums to get a sense of what is out there, what folks are liking, and what appeals to you best.
For more on batteries go here:
https://www.e-cigarette-forum.com/blogs/mooch.256958/
https://www.e-cigarette-forum.com/blog-entry/baditudes-blogs.7609/
Good luck!
Anna