If your guitar had 9's on it it is possible that the nut was cut to except 9's. 10's are bigger (stating the obvious i know) they might be a little tight in the slot which would cause them to stick and possibly snap at that point. You know if this is happening because the string will snap where it goes through the nut.
The remedy to this situation involes running a little bit of fine sandpaper through the slot to widen it a little, on the wound strings you can use a spare bit of the string as a file to widen them. Do not deepen the slot because the strings will be too low and buzz on the frets if you do it wrong.
I like to have my windings going from 2 on the post for the low E to about 6 on the post for the high e with the others somewhere in between. It doesnt always happen that way but i am usually close, it just takes practice. They should be neat and even without kinks or overlaps..
When restringing i usually put the string though the post (or in it on bass style tuners) and do one wrap by hand over the top of the hole to start me off. My thumb holds the string close to the headstock to make sure the rest of the windings go underneath the first and the other fingers hold the string taught above the nut whilst i roughly tune. Keeping tension on the string as you tune gives you much more even windings and therefore less tuning problems
