My BC puppy(10.5wks) will do this (throwing a fit) if I ignore him and he can't amuse himself because all his toys are "old." He has a lot of toys in a big bag that I switch out every few days so he has about 5-6 (2-3 chews, 2-3 squeaky toys), and they are all "new" to him. I only let him keep one or two if they are like his absolute favorites(these don't count towards his 5-6). After 2 or 3 days he is bored of them and I switch out the ones he doesn't want anymore with "new" toys he hasn't seen in a while. This keeps him from chewing on my stuff & eating my furniture.

He sometimes won't chew on his toys unless I make it seem like he isn't supposed to (like playing with the toy myself and "accidentally" dropping it, or playing tug and "accidentally" losing). He goes nuts when he thinks he's not supposed to have the toy.

Also, I second the ignoring thing. Ignore but make sure they have something to do so they aren't too sad. If you ignore but they have nothing else to do they will just annoy the heck out of you and maybe destroy stuff (depending on the dog).

Teach a "drop it" and/or "leave it" command. After they drop it, pick it up and put it somewhere else (if you can, if not just say"no" every time they go near it if it's furniture or something, I use "leave it" for when they bother my cats/other pets). BCs are smart, my older dog knows like 20 versions of the word "no" and more for "chill/relax", and "leave it/drop it". BCs are also sensitive to tone of voice, so if you talk in a stern voice it should be enough to make them understand you're upset (and that they should stop). My dog is guilty of ignoring me when I don't sound too mad.

If you have/use Facebook, there is this group called Border-Collie-Enthusiasts, where everyone is a BC owner and they all give great advice. Usually there's at least 2-3 people who have dealt with any specific BC problem that can chime in. They are overall pretty friendly people.