"You (are) damn right" is a way of saying "you are absolutely right". So the sentences means "you can be sure you won't" Since it's said with an authority voice tone, it's meant to say that what the first speaker said ("I won't tell anyone") is obvious for the second speaker. The second speaker also said it in a way that can be read as threatening --> "of course you won't (tell anyone) *unless* etc"

People throw "bad words" into spoken English to express anger, or hostility, or irritation. The "meaning" is "you won't--or I will do something bad to you" + "I am angry at you for talking about it." When bad words are used in this way, they do not carry their literal meaning. The speaker is trying to express the following idea: "I am so angry that I am out of control. I am not being careful to speak politely." In the 1940s and the 1950s, words like "hell" and "damn" were considered VERY bad language. Things have moved on, but they still express anger. "Damn" is bad language because it is related to the word "condemn." The literal sense of the word "damned" means "condemned to go to hell in the afterlife." "Go to hell!" or "damn you!" express the idea literally. But "Oh hell! I just hit my damned thumb with the damned hammer" just mean "I hit my thumb with the hammer" + "I am angry."

If they're saying "you damn right", they are forgetting "are" in "you are", which is long for "you're" , which native speakers confuse for "your". But "damn" is just an extra word for emphasis. It means the speaker is authoritative and angry at something.