Learn English – clutching one’s chest / at one’s chest

Either one is fine. Close to the vest has a more British feel to me, but I've heard both in the U.S.

EDIT

OK, since this apparently bothers someone, I did some research and it appears that the "vest" usage is more American. Note that my original statement of "British feel" was admittedly idiosyncratic.

Note also that either one is still fine, despite any individual's peevish disapproval.

Here are some NGram searches and their results:

British English:

American English:

Combined British/American:

Apparently the "vest" version came into British English in the late 1950s, but didn't gain widespread acceptance until the 1990s. Currently it looks poised to gain equivalence with the "chest" version, although such things are hard to predict.

Disclaimer I am not a fan of Google NGrams, because they can be used without regard for proper statistical practices. It can be hard to "clean" queries enough to be useful. The phrase "close to the chest" may, in fact, be over-represented here due to medical and other bodily associations, so I would expect some blue-line inflation in the above graphs. Similarly, "close to the vest" almost certainly appears in some references to garment making and wearing. Those things aside, I believe the "close to the chest/vest" idiom undoubtedly furnishes the majority of usage instances in each case.

"Break off" has two meanings that I can think of off the top of my head. The one that you refer to in your question can simply be replaced by "cancel" in all instances.

He broke off our plans <-> He canceled our plans

The second "break off" is a more literal phrase and it implies that something was removed from a greater whole.

"He broke off from the rest of the team." means he left the rest of the team and went off on his own. Likewise, "He accidentally broke off a piece of the vase." means part of the vase was removed from the vase.