I don't think there's any direct mention of Han smuggling spice for Jabba in the movies, it's just been a long held assumption at this point.
The way Lucas originally described how Han & Chewie operate, it's mostly entrepreneurial. They're mostly smuggling relatively small high value items that aren't illegal so much as they have very high Imperial tariffs. Basically, black market goods and it could be anything from luxury goods to engine parts to basic food rations for worlds under embargo. It all depends on what goods he can acquire cheap in one place that are in high demand elsewhere and get them in around local and Imperial customs checks.
Incidentally, this is why he's always either stinking rich one minute and perilously in debt the next. What he doesn't spend on the Falcon goes straight into his next shipment, which could end up being a total loss of the buyer refuses to pay, has been arrested when he gets there or tries to kill him.
What seemed to happen with Jabba is that the Hutt hired Han to move his goods (maybe spice, maybe not) from A-to-B, then return with the buyer's payment. When he was boarded along the way and had to ditch the goods, meaning he couldn't make the delivery (which doubtlessly annoyed Jabba's buyer) and thus couldn't bring Jabba his payment (which annoyed Jabba) and was out whatever his own fee was going to be, plus expenses (which probably really annoyed Han & Chewie.) Which just illustrates what seems like easy money can end up costing him more than he could afford and with no cash to invest in new stock, no clear way to make good, hence resorting to taking charters.
Would he have shipped spice? Probably, if the money is good. He may draw the line at death sticks, but even then maybe not. Given his background he probably figures beings are entitled to put whatever they want into their own bodies. A slave cargo is probably the only kind he'd flatly refuse to touch on strictly moral grounds. Also, Chewie rip him in half at the mere suggestion.