Do engagement rings have anything to do with Judaism?
In Biblical and Talmudic times, a couple would first have "erusin", a first stage of marriage; but would wait a year before living in the same house (the second stage, known as "nisuin.")
Today, the delay between erusin [giving the ring] and nisuin is about ten minutes -- it's the time when they read the Ketubah under the chupa.
Halachic erusin is much, much stronger than an "engagement"; it requires a full-blown halachic divorce to end it, and violating it is considered adultery. Nonetheless, modern Hebrew has started using the terms "arus"/"arusa" to refer to an engaged couple.
There are certainly halachos on the books about at what point a couple has either committed, or even sworn, to marry each other; breaking that contract without the other's permission is a very serious problem.
But as far as an engagement ring per se: I see no reason why they would be a chukas hagoyim problem, or any other problem, if the couple wants one. (I don't see why it would be required if the couple didn't want one either, come to think of it.)
60 years ago in America, they'd have these engagement parties at which the guy would publicly present the girl with a ring. Rabbi Yosef Eliyahu Henkin discussed whether this constituted full-blown erusin. Today, the ring is usually not given with a declaration in front of witnesses, so I don't think that's a problem.
1The shaila about engagement presents constituting erusin can be found in kidushin 50b– YDKJun 24, 2010 at 3:34
The Lubavitcher Rebbe was strongly against giving a ring as a present to a bride before the wedding. HaNesu'in Kehilchasam 3:97 is reference, but I don't have the sefer. see here: sichosinenglish.org/books/eternal-joy-2/04.htm#t4– MenachemJun 6, 2011 at 22:39
@Menachem (or Alex, or anyone else in-tune with common Chabad-Lubavitch practice) -- is standard Chabad-Lubavitch practice today to have an engagement ring?– ShalomJun 7, 2011 at 13:51
2the ring is usually given sometime after the chupah. I don't know if it is a universal custom, but many give it in the Yichud Room.– MenachemJun 7, 2011 at 18:53