Did the american "chick" come from mexican 'chica'?

Question

and why do the british call girls 'birds'?

Answer 1

It is far more likely that the American chick came from chicken.  In Spanish, chico/a means small and, by extension, it applies to young people. I can't tell you why the Brits call young women birds, but it is consistent with a term such as "hen party."

Answer 2

in searching for the etymology for 'bird' I found ( in etymonline.com):

bird, meaning young woman probably from 'burd' -

burd  

poetic word for "woman, lady" in old ballads; later "young lady, maiden," c.1205, perhaps from O.E. byrde "wealthy, well-born." Or a metathesis of bryd "bride." The masculine equivalent was berne.

I agree that chick is shortened from chicken.  Women are often referred to as 'duck' or 'duckie', particularly in the North of England.