Sir Percy, also known as 'the Scarlet Pimpernel' to the reader at least, is compared to all of these creatures in different ways as a sign of his bravery, slyness and, in the case of Chauvelin, as a sign that others for a long time do not know of his identity as the pimpernel.
The eagle is associated with power, freedom, courage and strength and was a well known heraldic symbol at the time and, indeed, the American eagle is still used to symbolise many of these virtues to this day.
Equally, one could argue that the fox is symbolic of slyness and cunning. Of course, in a particularly English setting - Sir Percy being a representative of Englishness during the French revolution - the fox was also recreationally hunted by the English upper classes. Sir Percy, in the guise of the Scarlet Pimpernel, is also hunted and evades capture, like the fox, as a result of his cunning.
The comparison to Chauvelin is a more complex one in so much as the two characters are opposed within the novel. The fact that both are utterly convinced of the correctness of their positions might make them comparable, just as Chauvelin was formerly acquainted with Sir Percy's wife and, in that sense, they are also comparable.