No, this word is not like imply cowardice in English. In Vietnamese, if you say someone like chicken "gà" it means this person always think for the problem or something very simple, not understand the reality of the problem. This word can also mean the skill of that person to solve the something problem is very bad, not enough to resolve. So it is only used for your best friends, it is not polite when said that to other people.
In the grammar notes, it indicates that "không phả là" is used in the case of identity, and with the "to be" verb. In this case, there is a negation of the identity of being a chicken.
In other cases, "không" is used without "phải là" in all other cases (that are non-identity).