how do I lay cement board on a concrete floor with out screws?
Why do you want to? If you're laying tile, just use a thinset made to bond to cement.
That would be an improper installation. The screws push the board down into the adhesive. Go to the manufactures website. They have guides on how to properly install cement board. I used Durock to build a hearth for my wood stove. Their website actually had an installation document on how to do it.
If the concrete floor is smooth, you can place the tiles on the concrete. The thinset will do the job. If the concrete is rough, try concrete refinisher. It's easy to use and will give a smooth surface. Hardi backer or cement board should be screwed to the floor. Use a masonry bit to drill the holes. Be sure to tape the seams and then proceed with your tile. Using a high speed drill will drill the holes quickly and easily. I do this for a living.
look for a web site for the product your buying or ask for an installation sheet at the store. It will tell you what you can use and how to do it.. Type of glue etc. you don't want it to pop up and leave a springy section in the floor that bounces when walked on. I would think if you glued it down you would have to weight it down overnight.