As of October 2014, break-inside still seems to be buggy in Firefox and IE 10-11. However, adding overflow: hidden to the element, along with the break-inside: avoid, seems to make it work in Firefox and IE 10-11. I am currently using:
overflow: hidden; /* Fix for firefox and IE 10-11 */
-webkit-column-break-inside: avoid; /* Chrome, Safari, Opera */
page-break-inside: avoid; /* Firefox */
break-inside: avoid; /* IE 10+ */
break-inside: avoid-column;