The breath of the unpatriotic man reflects the fact that he may be alive in body, but not in soul. Indeed, without a passionate love for his country, he has no soul; all he cares about is himself, his wealth and power. When he dies, he will go unlamented to the grave; no songs will be sung in his name:

For him no Minstrel raptures swell;
High though his titles, proud his name,
Boundless his wealth as wish can claim;
Despite those titles, power, and pelf,
The wretch, concentred all in self,
Living, shall forfeit fair renown,
And, doubly dying, shall go down
To the vile dust from whence he sprung,
Unwept, unhonour'd, and unsung.

It's interesting that the Greek word for breath, pneuma, is also used in a religious context to mean "soul." In "Patriotism," Scott is contrasting the soul's breath, which he equates with patriotism, with the act of respiration in a physiological sense. The unpatriotic man can certainly breathe in the latter sense, but then so can dogs and other animals. Scott appears to be suggesting that there's something less than fully human about a man who doesn't have love for his country.