I think what you need to do is to practice Slow Talk by yourself first and/or with someone you feel comfortable telling what you're doing. That way it becomes more natural when you're in an anxiety inducing situation. I remember Dr. Richards mentions something like this, too.
Regarding the reference to "ANT's," this idea will come up again and again, so why not shorten the name into an acronym to make it more concise, right? Similar to saying "SA" instead of "social anxiety."
Personally, the name "ANTs" gives me a mental image of annoying little pests that I need to get rid of or else they'll keep on destroying my life.
My two cents after going through most of the program.