If the "God Particle" was ever a thing, it would be the Higgs boson, and we already discovered that. It doesn't (and indeed can't) explain the conditions at the Big Bang, which are altogether more complicated sadly!
To answer some of Rev green's suggestions, although bear in mind that there's a good deal of "this is how it is so maybe that's how it always has to be" implicit in what follows:
// Who needs the neutron, except people doing carbon dating? //
The neutron has a vital role in keeping nuclei together by virtue of balancing the protons. The strong force holds all nucleons together, but has to act against the natural tendency of like charges to repel each other. By adding neutrons you get some "free" strong attraction without adding any electric repulsion. So it's probably at least harder to imagine a universe being interesting without neutrons. Besides, good look having a composite particle made of two up quarks and a down quark whilst excluding a particle with two downs and an up.
// We'd need photons to hold atoms together... //
photons can't really do that.
// ... W or Z to hold nuclei together...//
If anything, they do the exact opposite, since W's are responsible for changes of quark flavour in radioactive beta decay.
//... gravitons to hold stars together//
Gravitons are only really relevant in extreme gravitational situations, like collisions of black holes or whatever the hell happens near the centres of those things. You could get by perhaps without a quantum of gravity and just build a universe with basic Newtonian gravity, which perhaps is what you had in mind?