Assuming that you are logged into mIRC on your Twitch account, and that you are using another way to chat in the channel that is not the same copy of mIRC that this script it on, this will do the trick. Every time that you type anything in chat, your login on mIRC will try to change your color for you.

Code:
ON *:TEXT:*:#: {
  IF ($nick == $me) {
    VAR %hex 0 1 2 3 4 5 6 7 8 9 A B C D E F
    MSG $chan .color $chr(35) $+ $gettok(%hex,$rand(1,16),32) $+ $gettok(%hex,$rand(1,16),32) $+ $gettok(%hex,$rand(1,16),32) $+ $gettok(%hex,$rand(1,16),32) $+ $gettok(%hex,$rand(1,16),32) $+ $gettok(%hex,$rand(1,16),32)
  }
}