Can I put Hardie Board siding directly over old stucco?
For weathering in you home, it would be best to remove the stucco.

To add strapping, then the siding would place a lot of emphasis on how the windows are treated for weatherproofing. To do that properly, the stucco needs to be removed from around the windows and add thick trim and head flashing before the siding is set so the trim bridges the gap created by the strapping. 

To remove the stucco, it can be removed in sheets, really heavy sheets, really heavy sheets with a bunch of nails sticking out the back side that are in really solid... with wire teeth all around the edges ready to rake the skin off of you. But doing it this way using gloves, will keep the trash to a minimum.. If a right angle grinder is used to cut the stucco, then the wire ends will be kept to a minimum.

Real stucco is 3/4" thick, and solid masonry, with expanded metal lath reinforcement. The stucco can be thicker, or thinner, but 3/4" is the norm. If you go that way, as in removing the stucco, it can be elaborated on farther.

More on adding it over the stucco. It would be best to find the studs to attach the strapping to, since when the siding is done, there will be a lot of weight hanging on each piece.