Battery start for generator
Not all of those "portable car starting box" products are suitable for being permanently connected to a system with a trickle charger, which is what you would have to do. The batteries will cease to accept a charge if not sufficiently discharged periodically. So you need a system that can be on a trickle charger all of the time to be ready for action when called upon. 

Normally on a piece of equipment that is used on a regular basis, like a lawn mower or tractor, the starting battery is re-charged by the alternator once the engine is running. But on a backup generator you don't run the engine often enough (you SHOULD test it monthly, but hardly anyone does...). Hence the need for the trickle charger to keep the battery ready to go. That then also adds the necessity for periodic battery maintenance as well, so keep that in mind. These are the reasons why a lot of backup generators are pull-start; all you need to start them is some Wheaties.