"It's not clear to me whether they support Wallace or not, the boo's could be sarcastic or they could be imitating protestors. I think the main thing they're saying is that it's not their fault or their problem. That's why in the next line they say "Now we all did what we could do". Then the following line is basically saying, "If Watergate doesn't bother me, why should Wallace?" I think more than making a specific point it's just trying to be irreverent about politics while talking about how great the south is."
I suppose you could interpret it how you want, but it's clear to me that what they mean is:
"In Birmingham, they love the governor (boo! boo! boo!)"
- Birmingham was a hot bed of racism, they loved the governor. The boo! Boo! Boo! is a protest to that.
"Now we all did what we could do"
- Birmingham was at the center of the civil rights movement
"Now Watergate does not bother me"
- I have far worse things to get bother about. I've not judging you because of Nixon.
"Does your conscience bother you? Tell the truth."
- You're a notherner, Nixon is your prez. Why are you judging all southerns because of Wallace. I'm not judging you because of Nixon. How can you be free of your conscience if you won't let me be free of mine.
The gist of the song is: "Alabama is my home. I love it. She ain't perfect, but we make the best of it we can. Don't judge people because of where they are born."
Note too, that Neil Young and Ronnie Van Zant were friends. The song is not a put down of Neil, but it is a rebuke of his generalization.