how to charge battery and connect battery to jayco eagle 10

Wrong type of battery. What you have is a battery for a car. You need a " deep cycle" type battery.

Also, the battery cables, under most circumstances, black should be "positive" & "white" negative. Just connect them to the correct corresponding battery pole.

In the pic for the converter, it appears there is a Toggle switch, marked "conv" , off , & batt. The conv = converter, shore power 120vac, batt = battery power . Just switch to what your selected power source will be.