As l'électeur notes, the emphatic ない in 大事【だいじ】ない, 切【せつ】ない, and the like comes from older なし, and is an adjectival suffix indicating "having XX quality or state", where XX is the preceding root word. In terms of meaning, this is definitely not the same nashi as 無し.
There is a growing body of research, including recent works by Bjarke Frellesvig, building on the idea that Old Japanese had a copular verb ぬ. The emphatic ない is probably related to this in some way -- a number of other old root verbs have counterpart adjectives following the same vowel-shift pattern, such as あく > あかい, たく > たかい, なぐ > ながい.
Negative ない appears to have grown out of negative ぬ somehow. Although the exact derivation is unclear, the resources I've looked at (Shogakukan, Daijirin, Daijisen, among others) all agree on this point. Negative ない as a verb ending appears as an eastern-Japanese term from the late Muromachi period.
Negative ぬ must follow the incomplete form (未然形【みぜんけい】) of a verb. If it follows the stem or compounding form (連用形【れんようけい】), it instead indicates completion or resulting state -- essentially the same as emphatic ない.
This leads me to think that the negative connotations of ない originally arose from the incomplete or inchoate senses inherent to the incomplete conjugation of verbs. In my own research, it thus appears that negative ない and emphatic ない both arise from this same ぬ copula in Old Japanese.