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Very nearly. The mass of the proton is 938 MeV; the neutron is 939 MeV. And the physics at a proton-neutron or neutron-neutron collider would be very similar to that at a proton-proton collider. But neutrons are neutral, as you and others have pointed out, and therefore much more difficult to accelerate.

Now you could imagine a collider with a stationary neutron target and a high-energy proton beam. But remember that what you get out depends on the energy as measured in the center-of-mass frame of the colliding particles. To reach the LHC design energy of 14 TeV, you can collide two protons, each with an energy of 7 TeV in the lab frame, or you can collide a neutron at rest and a proton with an energy of ... excuse me while I dig out my TI-85 ... 104 PeV. Holy cow. I don't think anyone here has any idea how to get a 100-PeV beam in a working collider experiment, and I'm sure we don't have the money. So protons it is.

I think there's a misconception here. Protons (and neutrons) don't "contain" Higgs bosons, or W and Z bosons, or top quarks, or high-pT jets, or any of the other interesting things that we see at the Tevatron and will see at the LHC. These things are created from the kinetic energy of the two colliding protons. But otherwise yes, if you could find a way to build a neutron collider, you'd see pretty much the same stuff as at a proton collider of the same energy.

Oh, and I must rant:

Please don't call it the "God particle". This unfortunate nickname was coined as a marketing ploy and is not apt. Physicists do not call it the God particle. Reporters call it the God particle. And the main result is that people become confused, frightened, or angry.

I'm tempted to point out that if you're interested in a theory describing the universe we happen to live in, the Higgs boson is far more likely to be relevant than string theory. But maybe I should leave that discussion for another thread.