Probably not - the wiring for the adaptor is small, and cannot handle the load well from the jump. Your fan may not even run, while you may blow fuses. Better is to connect the jump directly to the main battery itself, then it would use the intended wiring and everything would continue to work.
But not for long. The battery jump starter is nothing but a small battery, so it won't have much capacity and would likely give up relatively soon after you plugged it in. You should figure out the amp hours of your current battery, the jump battery, and how much you need each hour to run the furnace, to see if the total capacities meet your needs. Don't forget, it is generally bad to use more than 1/2 of the total amps a battery is capable of.....
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2012 Ford F-150 Eco, SCrew, 4x4, MaxTow, HD Payload
SOB, used to be:Jayco 26.5 RLS
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