I once brined, then dry rubbed a pork shoulder (not a PICNIC one) and made some great smoked pulled pork. So brining AND rub is no problemo.
Why you want to use a picnic ham is beyond me, unless it's just that it's so cheap.
I once saw them on sale way cheap so looked them up from Good Eats. In episode Q (the pulled pork one) he said you need something fattier than a loin, but the picnic ham is too gristly. He went for the shoulder, which indeed makes fine pulled pork. In his Ham episode, he did mention you can make a ham-like product, but not a real ham, out of a picnic shoulder.
I know Cuban roast pork is more of a loin thing, but I wouldn't try to go with a leaner cut for smoking and pulling.
Nancy