Short Version : Not Animals, not plants.
could animals evolve sometime during the Hadean Eon, 4.6-4 billion years ago? Or would something hold them back?
We have only one example of the evolution of life to go on, so absolute statements are hard to make.
However ...
The first life is thought to have appeared in the Archean Era and the most direct evidence for that is for Stromatolites about 3.5 billion years ago, which are due to single celled microbes.
To get to animals, which is what you ask about, took until the Cambrian Era which was "a mere" 540 million years ago.
So based on our one example of evolution, it takes about 3 billion years from life to go from single cells to animals (and we're including very basic sea creatures in this loose definition of animal).
So if you start your evolutionary cycle in a Hadean Era, it's only about half a billion years before we think the first life appeared on Earth. So you shave about half a billion years off the time when animals appear.
Or would something hold them back?
This is an opinion, rather than something I can rigorously support, but I would suggest that while life based on the chemistry possible in the Hadean Era may be possible (I can't say it isn't does not mean I think it is), life based on the more stable geological conditions and different chemistry of the later Archean period may simply be better able to flourish.
Also the change of conditions might simply make it impossible for the multicelluar organisms that would (optimistically) be in existence at the end of the Hadean Era to continue. It's like any life existing in the Hadean would simply not be compatible with the conditions that developed later.
During the Hadean Era surface temperature is thought to be about $230^\text{o}\,C$ and atmospheric pressure was of the order of 27 atmospheres. By the Archean Era conditions have changed to (again very rough estimates) less than $50^\text{o}\,C$. As that era developed temperatures fell.
The issue here (and it woudl apply to any planetary formation) is that at the start conditions change (relatively) quickly. This will mean that anything adapted to the relatively extreme conditions of the earliest period will almost certainly not be compatible with later period.
Finally there is evidence of a period called the Late Heavy Bombardment which would have occurred at about the late Hadean-Early Archean period. This may or may not have affected the development of life on Earth, so it's a potential source of "wiping the slate clean". However I'd suggest (opinion) that what little evidence there is does not seem to support a break in the chain of evolution like this. It seems more likely that Hadean life forms (cells) would have evolved or died out as conditions changed due to the way the Earth's general state changed.