This is controlled by the Dimension Style. It is off by default in ANSI. To turn it on, you go to Styles, Dimension Style, and then the Units page. There is a check mark for leading zero.
You need to save this in your template if you want it to work for all new files.
Ah ha. I was getting ready to say, nope, tried that but it turns out I was modifying the wrong style.
It would help if selecting "Styles in use" actually worked...