You cannot really know what would the color of a chameleon be in a mirrored box, because you have to look through the box to know. Therefore - when you look through that box, the chameleon will automatically change her color to the color of your skin !
A chameleon doesn't really change colors abruptly, the chameleon changes colors slowly in order to adapt to its surroundings. If you put a chameleon in a room full of mirrors, it would shift its color to reflect the table it was on.
No, a chameleon can't change colors. It can go from light green to dark green to brown, but that's changing shade, not color. If you see a chameleon that's green and in a green tree, he hasn't changed green because of the tree. He was born green, but lives in the tree because he is the same color.