Historically there are other reasons for the Blackhawks/Redwings rivalry. It goes back to ownership, and you would have to be pretty old or a student of hockey history to know about this.
"Blackhawks owner and founder Frederic McLaughlin died in 1944. His estate sold the team to a syndicate headed by longtime team president Bill Tobin. However, Tobin was only a puppet for Red Wings owner James E. Norris, who had been the Blackhawks' landlord since his 1936 purchase of Chicago Stadium. Ironically, Norris had made a bid for what became the Black Hawks in 1926, losing out to McLaughlin. For the next eight years, the Norris-Tobin ownership, as a rule, paid almost no attention to the Blackhawks. Nearly every trade made between Detroit and Chicago ended up being Red Wing heists. As a result, for the next several years, Chicago was the model of futility in the NHL. Between 1945 and 1958, they only made the playoffs twice."
This is a brief account, and taken from a Wiki, but it at least gives a basic account of what was going on at the time. The Red Wings won 4 Stanley Cups during this time period, due in no small part to the incestuous relationship between the owners of the Wings and Blackhawks.
While most Blackhawks fans today may not understand the ins and out of that I think it's entirely reasonable to suspect that the hatred has been passed down from our fathers/grandfathers etc. I knew about this as a little kid - my dad told me about it. As a young Hawks fan I really didnt care, and over the last 15 years or so its been overshadowed by our own ownership issues, but I've always looked at the Wings as the team I like to see the Hawks beat and this old idea is probably the reason because I don't have any other real fundamental reason to hate the Wings.
Anyway, not trying to troll your board. Just thought I would toss in some insight.
