I'm not sure what "unevenly yoked" rules you are talking about.
It's a theistic term. Personally, I find the metaphor absolutely hilarious, particularly considering that it's the theists who use it. It's comparing the believer to an ox/horse and the religion as the cart that they have to drag around. Unevenly ... as in Protestants and Catholics ... liberal Protestants and the whack-a-loons, like the Pentecostals or Southern Baptists.
The term tends to particularly be used by the more fundamentalist sects, in my experience, which seems like a big uh-oh, in this situation. If his girlfriend's church elders would have a problem with someone unevenly yoked, say like a Catholic, I can imagine how they'd feel about the yoke-less.
As for his title question, though? Hell no. When you're marrying someone that you have reservations about, who's pushing for marriage because of the deportation issues you mentioned ... all while she's hopped up on pregnancy hormones. I'd be very careful about it. I have no idea what the alimony laws are like in Canada, but if you make much money, it sounds like a good way for you to get screwed.
The church still uses the metaphor only in reference to religion, though. The work they're referring to is the religious obligation.
That's very much the sort of thing that the original poster's girlfriend's church was referring to.
And either way, my mockery still stands. Oxen are yoked for the purpose of doing work not to their benefit. It's used as a metaphor in several places in the Bible, always speaking of subservience.
I dont really care if the child is raised christian by her. I think the best way to hate the church is experiencing it. I just dont want to get married.
The child is his "fault"? Besides not necessarily being true, what a sad view you have of his situation.
"Do the right thing."? The young man is seeking advice on the best course of action. Marriage it NOT necessarily it. Marriage never has, and never will, be able force two parents to become a loving couple.