I know the last reply to this was like a month ago but I'm new and browsing. hehe.
I honestly don't think Joyce over reacted, I just think both she and Buffy handled a difficult situation badly. Like a few of you said, Buffy had been admitted to a mental institution before for going on about vampires, so understanably Joyce was gonna think this all crazy talk, that her daughter was having a relapse. I mean, if you seriously consider your teenager daughter (real, hypothetical or otherwise) suddenly saying to you 'yeah, i slay vampires' your first instinct would not be, in my opinion, to believe them.
And I know she had just seen a vampire dusted in front of her but Joyce is pretty much Denial Queen and mix that in with the shock and speed at which everything was happening and everything is going to be pretty confusing for her. I mean she does try, when she's listening in to Spike and Buffy conversation/confrontation.
Spike: Me and Dru for Angel.
Joyce: Angel your boyfriend?
...
Joyce: So you didn't kill that girl?
Buffy: Of course not
Joyce: Did she explode like that man out there?
Buffy: She was a Slayer Mom
Joyce: Like what you are...?
But her whole reaction is so totally like how a parent would typically react when a child comes out revealing they are gay. Case in point: "Honey are you sure your a vampire Slayer? I mean have you tried not being a Slayer?"
She was slowly losing control of a situation Joyce felt she desperately needed to take control of. Everything was spiralling off in all directions and fantasy and reality were starting to bleed together. We already know from 'School Hard' that Joyce places authority on ability not status as she says to Buffy 'Principal Snyder said you were a trouble maker and I could care less. I have a daughter who can take care of herself and thinks of others in a crisis. No matter what dumb teenage stuff you think you need to do, I'm gonna feel better knowing all that.' and despite Snyder being in charge she tells him to follow Buffy's orders as she knows what she's doing and even critisises him for not following them 'Snyder: And I say this is not happening! Joyce: Oh well then! I guess the danger's over'
So, when, in Becoming, Buffy shows how much ability and power she has Joyce also realizes on some subconscious level that she doesn't have that authority/control over her daughter like most parents have, so she tries to reason, she grasps at straws, she demands explanation but eventually has to resort to pulling rank 'if you walk out of this house, don't even think about coming back'. Buffy walking out, no matter how somberly, shows that she has the greater ability and so the power/authority. I don't believe Joyce meant truly what she said though. Like most of you have said, it was something said in the heat of the moment. And you can't blame her- when asking for an explanation all she gets told by her daughter is 'just have another drink'.
As for the blaming of Giles, it's easier than blaming herself and Giles stands for all that her daughter is forced to be hence the blameage. The two do try to make nice again though in Dead Man's Party when they start talking Art.