There is no single correct answer to this question. As with many things involving capitalization/punctuation and slogans/marketing material, there is significant creative license. I would make the determination based on the extent to which you are referring to the phrase (slogan) in context. Thus, for example:
Nike has a "just do it" attitude to business.
This implies that the attitude itself is about just doing it. This would be true whether it was Nike who had this attitude (given that Nike's slogan is "Just Do It") or someone else, e.g. Adidas or Reebok. The emphasis is on the role of the words/phrase as a modifier. It's up to the reader to identify that this happens to be a slogan (the quotation marks help, but don't confirm that it's a slogan).
Nike has a "Just Do It" attitude to business.
It's possible to interpret this similarly to the first example, but here the focus is much more on the fact that it's a slogan and that Nike is living its slogan. If we said "Adidas has a 'Just Do It' attitude to business," the reader would be led to infer that Adidas is very specifically taking a cue from Nike in some way.
Also, with this capitalization it's less necessary that the grammar works correctly, because it's clearly taken out of context. A different example will better illustrate what I mean:
- I agree that Disneyland is "the happiest place on earth."
- That "The Happiest Place on Earth" sign reminds me of Disneyland.
The capitalization in the second sentence helps because the quote is so clearly out of context of the grammar of the entire sentence. By contrast, in the first sentence, the slogan (a noun phrase beginning with an article) fits right in as the predicate nominative of the sentence, and the capitalization to set it apart is less necessary.
At the end of the day, however, you could be considered correct using either example. This is a stylistic choice at heart, not a purely grammatical one.