the breath of life is mentioned several times, not just in Genesis 2:7, and not just in Genesis:
God that made the world and all things therein, seeing that he is Lord of heaven and earth, dwelleth not in temples made with hands;
Neither is worshipped with men's hands, as though he needed any thing, seeing he giveth to all life, and breath, and all things
(Acts 17:24-25)
For behold, I will bring a flood of waters upon the earth to destroy all flesh in which is the breath of life under heaven. Everything that is on the earth shall die.
(Genesis 6:17)
as long as I have life within me,
the breath of God in my nostrils,
my lips will not say anything wicked,
and my tongue will not utter lies.
(Job 27:3-4)
My words declare the uprightness of my heart,
and what my lips know they speak sincerely.
The Spirit of God has made me,
and the breath of the Almighty gives me life.
(Job 33:3-4)
If he should set his heart to it
and gather to himself his spirit and his breath,
all flesh would perish together,
and man would return to dust.
(Job 34:14-15)
For I will not accuse you forever, and I will not always be angry;
for then the spirit would grow weak before Me,
even the breath of man, which I have made.
(Isaiah 57:16)
Thus says God, the LORD, who created the heavens and stretched them out,
who spread out the earth and what comes from it,
who gives breath to the people on it and spirit to those who walk in it:
(Isaiah 42:5)
But after the three and a half days the breath of life from God entered them, and they stood on their feet, and terror struck those who saw them.
(Revelation 11:11)
there may be even more than this - but certainly that's enough to see the truth?
you should be able to clearly see that the animus the breath of life, that that gives all that lives it's life -
this is not the same as the Holy Spirit.
i could show you many verses where a prophet prophesies, and it is said "the spirit of the Lord came upon him.. " or something similar - but the one the spirit of the Lord came on was already living and breathing and animated, and was also still alive when that spirit left him. just like with the disciples, who were already alive before pentecost, they had in them the breath of life, but they had not received the Holy Spirit.
now, the same Holy Spirit that we have may in fact be the same as the "Spirit of the Lord" as it was called in old times, that caused men and women to prophesy; that's not really clear in scripture - even in Revelation "the seven spirits of God" is mentioned (Revelation 4:5). but what gives earthly life to people is not the same spirit or breath as the "Comforter" that Christ promised to send to those that trust Him.
this simply cannot be the same as the Holy Spirit. if it were, then all men that live are full of the Holy Spirit, and that is certainly not true, for many (what should be) evident reasons.