Do Ukrainian speakers accent words differently depending on where they're from? I've heard most examples of дівчина sound like дів ЧИ на but here it sounds like ДІВ чина.
The word «дівчина» can be stressed either way (ді́вчина or дівчи́на) without change of the meaning. The printed version of «Академічний тлумачний словник (1970—1980)» only has «ді́вчина»: http://sum.in.ua/p/2/298/2 — however, in modern language both ways to stress it are acceptable, and the online version of the same dictionary, interestingly, marks both places of stress as acceptable: http://sum.in.ua/s/divchyna
Yes, the accent really depends on the dialect. People closer to the West are more likely to shift stress to the penultimate syllable, because of the Polish influence; and people closer to the East are more likely to be influenced by Russian. So, the accent «дівчи́на» might be an influence of Western dialects (although I can't vouch for this!).