Can this be done using a 556 ic?
No
It's crazy enough thinking of 10 second delays using a 555.
The accuracy, the sensitivity to leakage and change of the electrolytics will kill you.
You want to fine-tune for 10 seconds? You might need 10 cycles to get it straight, that is 10*(10 sec+10sec) = 3 minutes * 100 units = 5 hours
Oh, a 24 hour cycle? It will take you a week to adjust each unit.
Also, how would you put each unit in operation when you set it up at a random point in the 24 hour cycle?
If it were just to be on for X hours out of every 24, a mechanical timer would be the easiest, cheapest.
If it needs to cycle at a proscribed rate for X hours out of every 24, you had better use a microprocessor with a display so that you can set (and verify) the current time, the active times and the cycling parameters.
Also, this whole project is a moving target.
The next requirement will be that the unit shuts down earlier on Sunday and holidays.