The only things that break concentration are listed in the PHB, the PHB says on 203-204:
Normal activity, such as moving and attacking, doesn't
interfere with concentration. The following factors can
Casting another spell that requires concentration. ...
Taking damage. ...
Being incapacitated or killed. ...
Unless you fall asleep, you are able to maintain concentration on the spell. There has been some question on forums about the reverse, that is "Can you really rest if you are concentrating on a spell?"
The answer to this question seems to be yes also, as:
[S]hort rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
I'd argue reading would require a similar amount of focus as concentrating on a spell. The designers never commented on this aspect, but their posts about Hex make me believe you can indeed carry it through a rest, and benefit from the rest.
And Mearls answers this question on twitter, which indicates it is the developer's intention, even if it isn't an official RAW ruling:
Q: Can I start a short rest while keeping concentration on a spell like Hex?
A: as long as you don't sleep