L.,
This, to me, is concerning. One thing which really raised a red flag to me was that when you express your opinion to your husband, he says you are 'backtalking'. This isn't how most adults --who considered each other equals--would address each other. The import in the word 'back talk' is that one is defying or talking back to their superiors.
Please find a couples counselor who will help you both gain support. I agree with a previous suggestion of your husband getting help in managing his emotions. I'm admittedly not a huge supporter of spanking (I have spanked my own son, by the way), and I can tell you that being spanked by an angry parent in the heat of the moment while being yelled at is very different than a calm parent thinking "oh, boy, this is going to hurt me more than it's going to hurt you".
Parenting classes will help, too. It would be great if your kids could have two parents who are on the same page regarding discipline and who felt approachable.
I realize that lots of kids go through phases of saying "I don't like" one or the other parent, especially when their feelings are hurt or one parent is present more than the other. And when I first saw your post title, I really thought "Oh, over-concerned parent--this is pretty common". However, what you describe in your post sounds like the makings of a long-term power struggle between growing kids and their father and a bad way of relating to each other (and other people) in the long run for your boys. Please do find some help. Being told calmly "when you correct me in front of the kids, I feel you are undermining me, can we find a way to get together in how to deal with this" is very different from being told 'your place' in the relationship--which is the vibe I got from your post.
Best wishes going forward. It won't be easy, but it will be worth it.