The universe appears to have a finite age in the Big Bang model and can be calculated to give an approximate value. Particles however, do not seem to have any correlation with time. The lifespan of a fundamental particle, without colliding with its anti-particle through annihilation, appears to be infinite.
The universe in the Big Bang theory does have an age (approximately 13.7 billion years), so in a way, the Big Bang theory does not explain the origin of the universe, but rather the state of the universe 13.7 billion years ago. With that said, it is the limit of the universe, not the particles. The age of a particle could be infinity.
Yes, some of the elemental particles in our body (electrons, quarks, ...) were actually popped into existence along with the universe during the first fraction of time of the Big Bang, and remained still until today. So whatever or wherever these particles came from (assuming they do), they carried with them information that is older than the universe itself.