Hi! I believe I can be of help with this issue.
First, let me say I get so many questions stated so many ways, yet yours is truly unique!
You are clearly a person who ponders things and tries to have good and deep insights into yourself. And sometimes, it is the hardest to unlock the understanding of those things that are closest to us, that refer to us in meaningful ways deeply. So let's set about unlocking these words.
The key here is Linus's blanket. Charles Schultz created Linus as someone who ventures into areas of exploration that are very feeling based. This was represented by the piano. And when the world would swirl around him he would reach for his blanket. Or when Lucy would seek to establish a more intimate relationship with her fantasies of their future together, he would often reach for his blanket. So when things got too much for him, Linus's blanket was his security blanket. Meaning he withdrew inwards, he withdrew within himself, he withdrew from the word.
Based on that the "balls" here I suggest are referring to testicles. To "have balls" in American vernacular means to be brave, to have guts, to confront things in a forthright manner and decisively. It is the opposite of Linus's blanket: confronting the problems as opposed to withdrawing and seeking security. Being brave instead of seeking comfort.
This is my impression of what Dr. Phil's dad meant by this expression.
I wish you the very best!
Please remember to click the green accept button. Feel free to continue the discussion; my goal is to get you the best answers possible. Bonuses are always appreciated! If I can be of further help with any issue, just put "for Dr. Mark" in the front of your new question, and I'll be the one to answer it. All the best, XXXXX XXXXX