You would need to break the connection between the +ve of one battery and the -ve of the other when putting them in parallel from series. And then break the +ve to +ve and -ve to -ve when moving them back to series from parallel.

If you connected them as both parallel and series you would short the batteries out, consider this schematic

(Probably not the best schematic ever but it was rushed) battery 2 is shorted to ground, therefore the voltage at any load is 0v and the spectacular explosion coming from the batteries would be quite the sight.

You would need to create a switching circuit to change the circuit from parallel to series.

However, in this schematic (again, not great but will do);

With switch 1 open and switches 2 & 3 closed you have a parallel arrangement With switch 1 closed and switches 2 & 3 open you have a series arrangement

Unless I have misinterpreted your requirements and you want battery 1 to power EZ-B 1, battery 2 to power EZ-B2 and a link between the 2 batteries to make them series (i.e. 12v) for charging, like this;

Which still wouldn't work. EZ-B 1 gets 6v but EZ-B 2 gets nothing since the positive of the battery is shorted to the common ground.

Do away with the common ground in your drawing Rich. Each battery can provide power to each EZ-B and still be in series to provide 12vdc for the charger.

Can you do away with the common ground?

And wouldn't Battery 2 still be shorting to ground (as in EZ-B 1/Battery 1's ground)?

Don't think so. As each batteries leads are extended to each EZ-B only the 6 volt potential will be present on each board.

Another way to test this solution is to place a voltmeter at the connection points of each battery. You will measure 6 volts on each battery but will see 12 volts when measuring from Bat 1 (-) to Bat 2 (+).

So far as the ez-b boards each running off of its own 6v battery you should have no problem. The batteries can be then connected in series to provide 12 volts to other devices. There are two issues.

The first is that you want to charge the batteries while they are in series. In this configuration you will by its nature create a cell imbalance between the two batteries. Each ez-b board will drain its respective battery at a different rate from the other one. Depending on cell type most manufacturers have warnings about the effects of charging systems and imbalanced cell batteries. This should be a concern.

The other problem has to do with the grounding and control of the 12v components. Only the ez-b that shares the ground with the 12v devices should be used to control the 12v devices. The other ez-b with the floating ground should only control 6v devices which share its ground connection or devices powered by the ez-b with the floating ground.

Lots of chances for releasing blue smoke and potential for shorting out a 6v battery.

This would not be my first choice for a setup.

Agree with @RobertL184 about battery balance issues. Grounding issues and running 12 volt components may create issues that can be cause for concern. My initial discussion was more of a tunnel-vision of the initial question and a solution using two batteries in series that could each power an EZ-B.