How can I control a Non-Addressable rgb strip that is cut every meter and each piece to light up in a selected color?
Any small microcontroller an create the necessary PWM signals for the colors. A driver board will be required to drive the 3 ceparate colors - no µC can provide the necessary power.
This µC can also read the button state and do the communication.
Plan is make more than 600 pieces of each.
Do you mean 600 of these controller|button|strip units (let's call them "node") in series?
I²C can address up to 127 devices on a bus. Using 5 additional chip select lines, one for each block of 127 devices you can address up to 635 devices. But not at the distance you show (with 1 m distace between each unit the total is 600 m from the master controller to the last device).
I'm not familiar with DMX, but:
DMX would be suitable for longer distances (repeaters will be required). One DMX line can address up to 512 channels. Depending on the resolution of the light effects one node could work with one channel (256 different pre-programmed effects) or it could require 3 channels (one for each color). Therefore you may be able to do with one DMX line (if you can reduce the max. number of nodes from 600 to 512), or you may need 3 DMX lines.
Since DMX is used widely to control light fixtures you may be able to find suitable masters and slave nodes off the shelf.