Big Bang theory certainly doesn't support everything coming from nothing, since that wouldn't even go along with everything else we know about astrophysics and the like: that is, everything changes form, due to changes in energy and matter, that sort of thing
The collapsing in on itself theory seems to be unpopular among many astronomers, known as the Big Crunch theory. Not sure what the exact alternative is, but I'd guess it's more a notion of universal gravitation: the universe as a whole would eventually reach equilibrium and then everything would gravitate towards a single point and we'd repeat the big bang all over again, hypothetically speaking.
This assumes that there isn't another big bang after the universe is reduced to a singularity. Correct, time in this universe would stop, but that doesn't stop time from beginning in a new setup in the new universe. Days might be shorter, longer, who knows?
Not sure how this is supporting your argument that we should aspire to live forever, especially if we would eventually just become part of the quantum singularity that starts another big bang. Pretty lonely existence there.