Can the statement be regarded as true? That every particle, or element in the universe can be regarded as a combination of black hole and white hole in variable proportion.
A "White hole" is the time reverse of a black hole, as discovered by Hawking and explained by 'tHooft. The reason is that black holes can be in thermal equilibrium with radiation, and the time reverse of a thermal equilibrium state is still thermal equilibrium. This intuition is confirmed in AdS/CFT, where a thermal black hole in AdS can be described by a time-reversal invariant thermal boundary state, and so one is safe to conclude Hawking's argument is valid in quantum generality.
The question of whether particles are black holes is to my mind entirely answered by string theory. The first point is that black holes are classical, and you are talking about quantum particles, so the question is only whether the particles can be interpreted as black holes in an appropriate limit.
The F-strings from which the particles in the standard model are made are dual to branes, either by going to strong coupling if you are in a heterotic string theory, where the strings are revealed to be 2-branes wrapping a dimension, or else by going to strong coupling in IIB string theory, where you find that the F-string swaps with a D-brane.
In either case, M2 branes or D1 branes are both classically intepretable as black holes, because when you stack them up, their supergravity solution is exactly a black hole. So within string theory, it is not wrong to say that every particle is a particularly highly quantum black hole, in the smallest quantum in which it comes.
This doesn't mean the classical GR picture works for these objects, it doesn't. The other answers explain that it fails completely. But the quantum picture that all matter is extremal black holes (excluding orbifolds and other exotica) is the basic picture of modern string theory. It unifies particle physics and black hole physics in a fundamental way.
You first have to understand what a "white hole" is. It's the time reverse of a black hole. It was rightly pointed out in previous answers that white holes violate the second law of thermodynamics. Now, like anything in thermodynamics, this makes them unlikely but not impossible (unlikely here usually means unlikely even in an astronomical number of universes ...). But if you keep that aside, there is a more beautiful picture of what a white hole is: it is a quantum superposition of all possible black holes of about the same size. There are so many ways to make a black hole, and they come in so many microstates, that the quantum superposition corresponding to a white hole is astronomically unlikely, ... but not impossible.
Such considerations could be important if you ask what the relation is between elementary particles and black holes. This has been answered already: if you try to make such a comparison you would have to realise first that all known particles are so light that those black holes would be tiny, in fact billions of times smaller than the smallest distance conceivable in physics: the Planck length. Because of this, comparing known particles with black holes, or imagining them as being built from black holes, black, white or otherwise, is not considered to be a fruitful exercise in theoretical physics.
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Over the years there have been suggestions that elementary particles may be black holes. However no-one has ever been able to make this quantitative and I doubt anyone believes it these days. There was some discussion of this in what is the difference between a blackhole and a point particle, and Googling will find you lots of hits on this subject.
I've never heard of the idea that particles may be a combination of a black hole and a white hole. It's hard to see how such an arrangement could be possible.
The term "white hole" is widely used in science fiction, but SF has the wrong idea about them. The equations of general relativity are time symmetric, and if you reverse the direction of time the black hole solution turns into a white hole. However there is no evidence that the time reversed solution has any physical significance, and as the comments above suggest, no-one (in the mainstream physics community) believes that white holes exist.
The nearest we come go to a white hole existing are suggestions that the Big Bang was a white hole. Have a look at John Baez's article on this for more info.