Hi Dave,
I used the AutoZone free rental ball joint press, which I think is the same as the HF unit you have. I had to pull the dust boot off the joints and the joint body sat in the hole in the tool nicely, it did not press on anything inside the joint. Perhaps your press is slightly different in sizing, or your joints are different (mine were Lemforder).
Yes I did ALL steering linkage at the same time. I used a pickle fork, a cheap JC Whitney "lever type" separator, and a Bernz-O-Matic torch. The torch was the BIGGEST help - put the joint under tension with the tool, then apply heat - after 30-90 seconds you get a BANG! and the joint pops out.
I also did the guide rod support joints, those were SHOT, and basically every moving part in the front end. It also has new calipers, rotors, & brake hoses... handles MUCH nicer now!