What is the difference between 一只狗 and 一条狗?
There's no negative connotation, at all.

条 and 只 are used interchangeably in today's Chinese, not just specific to a region. Long ago, 头（頭）were used as a quantifier for dogs or other farm animals. I suspect "head" was dropped because dogs do not have stocky builds as other animals(pigs, donkeys, bulls etc). Also "头" tends to associate "dumb animals" and dogs are perceived as intelligent.

条 in most cases is used to quantify something that's slender, e.g. 一条路(road)， 一条河(river)，一条虫(worm). 

I say 一只狗 because it seems to make more sense considering others animals of simliar size and stature also use 只. e.g. 一只猫，一只狐狸 etc. But this is just a personal preference.