I've got a couple of Stick's, a V1 (well over a year old, still working, but with a dodgy button) and a V3. I've not heard of any particular problems of batteries exploding in them, but happy to read up on it if anyone has links. Needless to say neither or mine have blown up. Any lithium battery can vent/explode if mistreated though.
I would suggest using the batteries that come with it or like for like replacements. I bought some Ultrafire 10440's which claimed to be 600mah but I found them utter carp. They might have been faulty however. You might even try the new IMR10440's LiMn batteries which have a much better discharge rate than LiIon's, and will never blow up. This means they should support power hungry atomisers like 510's better. Here is one place that sells them, they are more expensive than other 10440's -
Generic IMR10440 3.7V Battery - Cloud 9 Vaping
I don't know of any batteries in that size that genuinely are a larger capacity. If larger capacity is needed you might be better off with an eGo or a mod of some type. Cheap box mods with 14500 batteries that have 2-3 x the battery capacity of the JS are very affordable.