Does input amperage matter in a DC-DC Booster?
Yes, certainly.  The output power will always be less than the input power, but specifically it will be a portion of the input power determined based on the converter efficiency (which is claimed to be 96% for that device).  So connecting two cells in parallel would allow approximately double the output current (assuming the converter can handle the current and doesn't limit it).

But that converter steps up to 5V.  If you are supplying 6V to it, the output may be up to 6V.  Also, that is not a charger, it is only a boost converter. You shouldn't connect it directly to a battery.