The existing Quote button next to each post more or less does similar job, just need to click once for every post.
The checkbox approach is nice, however it needs to do more to have the effort justified.
How does this sound:
Let's change the function of the current Quote button. Instead of opening a new reply with Quote, the new Quote button adds the quote to a queue similar to an online shopping cart. This should allow multiple quotes to be queued before a reply is compiled.
When the user is ready to post, he then hit the Reply button as usual. All the queued quotes in the "cart" will now be automatically included in the reply.
I think this "shopping cart" approach may be more attractive than using a checkbox. One reason being that the checkbox approach does not work very well for large threads with multiple pages. Current quote system also only displays limited number of posts available for quoting in the preview. At the moment the user has to open the thread in another window/tab to manually copy the link over in such scenario.