Opposition to gay marriage, anti-abortion stance, Christian-only symbols and such on public/government property, the "let's kill 'em quick" mentality of capital punishment proponents (those pesky Sixth and Eighth Amendments) ... Generally the only Constitutional right that is regularly and vocally championed by the Republicans I know, and I know a lot of them, is the right to bear arms. (Sometimes I hear about religion, but that is often because they feel that removing religion from school's official activities, which to my mind is perfectly reasonable since school is an extension of the government, is discriminating against them. I think that is just because no matter who you are, it's hard to have something taken from you that you've always had, even if not having that thing--in this case, primarily Christian symbols as an official part of public school--is the right way to be.)
Each of us has rights that we feel particularly passionate about. You will find some members of both major parties willing to restrict the rights that are not important to them. Since we are being honest here, I am going to say that although it is a *small* faction of the party--which you can see from ... some survey out there that I'll try to dig up if someone really wants it--the people who would like to really intensely cripple the Second Amendment are probably mostly Democrats. In other words, only a small minority of Democrats think no one should have guns, but the people who think no one should have guns are likely to be Democrats. Each party has its extremists. (I might be wrong. If most of the people who are totally anti-gun are actually Green Party or something, I apologize and freely admit that I just *assumed* it would be Democrats. But I know it's a minority of Americans, either way.)
Here's the difference, though: While I personally know Republicans (and/or Independents who often vote Republican) who do not agree with their party's stance on civil rights, abortion, etc., those stances ARE the face of the party right now.
The only ones I know who are really staunch about defending every possible liberty above everything else are Libertarians. And frankly, while I admire their adherence to principals, I find them overall to not be a particularly practical group of people. Their ideas would work okay in small groups where there's not much requirement for cooperation but are a little too "wild wild West" to be practical in society, I think.