I do think that basic jumping skills should be part of what goes into a well-broke horse, no matter what the discipline. It doesn't have to be formally "trained," I just like to know that the horse can mentally handle the obstacle. The horse should know that a jump is just another type of obstacle to be negotiated, like a bridge or a trailer or a gate, and be handy enough to pop over it without balking, running out, refusing, or bolting upon landing. It doesn't have to be A-Rated hunter pretty, or GP jumping high, but it needs to be willing and safe.
It doesn't bother me if my horse doesn't have this skill, I just add it to the list of stuff to eventually teach/introduce to him as he becomes ready. I generally lump neckreining into the same category as jumping - something that a well-broke horse should have a working understanding of, no matter what his focus discipline is.
As far as your horse, if he's green at jumping he will probably benefit from an appropriate jumping saddle. The dressage saddle just isn't designed to help the horse over fences, and a greenie needs all the help he can get from appropriate tack that helps his rider balance. Sure, it's possible to pop over a jump in a different style of tack, but it isn't going to make the process easy for a learning horse or rider. If you want to formally teach him to jump, and you aren't a very experienced jumper/trainer, I highly
advise getting some knowledgeable help to get your horse started over fences. The process can get hairy pretty fast.