Whatever you do, leave as many strings on the instrument as possible. Taking too many off can cause your post to collapse, and then you have to pay to have someone reach in there and give a post adjustment. That really stinks.
I think it should be ok as long as you are careful not to joggle it too much and are quick in getting the next string.
As to why it keeps happening, when you tune your instrument, do you use the pegs or the fine tuners? Perhaps tightening too fast will cause your string to snap. And are you sure that you have the right size of string? One more thing that seems likely (maybe most likely?) is the width of the string.
(It is a funny thing indeed to note that this not only had nothing to do with the problem, it doesn't even exist as a problem... ;-D)
Thicker strings require more tension to keep up to tune, as said here:
Almost all strings are available in different thickness or gauges, for example Thomastik Dominants, which are available in stark (thick), mittel (medium), and weich (thin). Pirastro Eudoxa, Olive and Kaplan Golden Spiral gut strings come in a variety of gauges indicated by gauge numbers. The majority of string players use the medium gauges. In general a thicker than normal string will require more tension in order to bring it up to pitch.
Consider slightly thinner strings. I use medium, so you know. Hope you find out what's happening, and do tell us when you do! Happy playing!