The Chromagen Eternity 26 is an on-demand hot water heater. Since it does not store water, it does not normally run except when you use water. If everything is connected properly, it also won't run in normal use in your case as long as the storage tank below it has sufficient solar-heated water to meet your needs. If the storage tank water is too cold (either because the solar is just not heating the water enough, or if you have used up the hot water and now the tank has cold water), then the Chromagen will provide additional heat as needed.
The Chromagen has electronic ignition. There is no pilot light wasting gas when you aren't using it. The control board uses a little bit of electricity to monitor the temperature & water flow, even if the system is not currently being asked to heat water. The manual specifically advises against disconnecting electricity & gas because if you do so and there is a freeze then the unit may be damaged, and that damage would not be covered by the warranty. Even if the warranty has expired, I think this is very good advice. If the temperature stays above freezing then you do not save any gas and only save a little bit of electricity by leaving the unit running normally. On the other hand, if there is a freeze then the little bit of gas needed to run the system periodically will far outweigh the cost of fixing the unit after pipes crack.
Leave it connected normally. You can turn any relevant thermostats down to the minimum, but otherwise leave it alone.
If you turn off electricity & gas for the entire house, the unit will be OK, as long as you don't have a freeze. In my area (Maryland, USA) it is normal practice to leave the furnace & water heater on at their lowest settings when going away in the winter, just in case.