Likewise in the Deep South, the white vote went massively GOP and I assume that includes those in unions.
There aren't that many union members in the Deep South anymore, because of the violent anti-union thuggery there.
True there are not a lot, but even of the one's there are I am pretty sure Romney handidly won them. Lets remember not all union members vote the way their leadership tells them to. Many nowadays have above average salaries so someone who wants lower taxes would appeal to them. Also the idea of smaller government is a view widely held by many Americans across the country from a wide variety of groups so running a platform of bigger government and making the US more like Canada and Europe it is not something that will win you many votes even amongst unionized workers. Most union workers are quite patriotic too and want to maintain American values.
This analysis is rather simplistic. Union membership tends to be one of the few factors measured to moify the voting habits of whites (especially white males). No, it doesn't work en masse, obviously, and I'd still expect a white service union member in Alabama to more likely vote Republican vs. a white member of the same union in Vermont for a variety of cultural reasons. But push comes to shove the absense of strong unions are a notable factor in low Democratic voters among Southern whites (though by no means the only, or arguably even the largest reason).