I'll second what MON said also

And I'll add - six years old was an age that each of my kids went through some personality challenges! From what I understand about child development (and I'm on expert on this) - 6-7 years old is a time developmentally when children are starting to see themselves as independent from their parents in very real ways for the first time, so it's one of the times in their lives where they are prone to being overly emotional etc. (Kinda like a test run for puberty lol). It also coincides with a time when they are typically in the first years of full-day school, first years of having to sit in a classroom, things like that, so their world has changed and they may act out more because of that.
I think that (just for me) sometimes it helps to not assume everything that happens with my kids is due to giftedness - I think this is a very normal time of life for all children to potentially have some challenging behaviors.
Hang in there - I also found with all three of my kiddos, that the challenging behaviors of 6-7 years old were a phase. It was still important to deal with them, to teach our family values etc, but in the thick of it it seemed very difficult to get our values across to them - but in the long run, the behaviors did mellow out and my kids morphed back into reasonable human beings (until they hit puberty!).
polarbear
ps - not sure this will help, but one very simple thing that I think has helped us when our kids were acting out or behaving in ways we didn't want them to was to talk to them one-on-one and ask a *lot* of questions about what was going on with them - when they were that age in particular, sometimes the behaviors were due to something else entirely than the situation they were in when the behaviors occurred.