>For example, the gene for eye color exists in every cell of the body, but the process that produces the protein for eye color only occurs during a specific stage of development and only in the cells that constitute the colored portion of the eyes. That information is not stored in the DNA.
Edit: I have a feeling that this article isn’t telling the researcher’s whole story. The implications of epigenetics are vast.
The concept of genetic canalization addresses the extent to which an organism's phenotype allows conclusions about its genotype. A phenotype is said to be canalized if mutations (changes in the genome) do not noticeably affect the physical properties of the organism.
This means that a canalized phenotype may form from a large variety of different genotypes. If canalization is not present, small changes in the genome have an immediate effect on the phenotype that develops. 
Mary Jane West-Eberhard famously told "“The 20th century has been called the century of the gene, the 21st century promises to be the century of the environment.” ... ” Darwin, who didn’t even know genes existed, had it right, he left open the possibility that new traits could arise because of environmental influence." 
An example of such epigenetic and heritable trait changed in morphology without change in DNA is here 
Most people would say that's because everything required to make a tooth- the process of making a generic tooth, not the structure of a specific tooth- is encoded in DNA. Of course, it also requires that there be various proteins present in the cancerous germ cell- cellular components that themselves are encoded by DNA.
If I were to put this as succinctly as possible, I woudl say "there appears to be some amount of non-DNA cellular state which affects development, that cannot be traced back directly to expression of protein from DNA in the cell, that is heritable when cells divide". Good luck proving that! The experiments would be nearly impossible to run.
People have proposed taking genomes and replacing all the parts that aren't under functional selection with random DNA, and showing the organism is still viable. Other folks have proposed cutting out those parts. Nobody has run a convincing experiment on a higher animal to show this is possible, or that the results are convincing in any way.
A quick concrete example in humans is the inheritance of antibodies between mother and child which is transfered via the placenta.
This is all the phys.org press release seems to be saying: heredity is DNA + all the other stuff. As far as I am aware, this is not controversial at all in biology. The actual substance of the two papers is a model for understanding that process, if it's a good model I am sure it would be a good contribution to evolutionary science.
If I am not mistaken, there are several evolutionary explanations, such as for a 50-50 sex ratio being a stable equilibrium, and for the benefits of altruism, that assume an organism's fitness comes equally from the male and female parent. If, however, the female parent has significantly more contribution to the offspring's fitness, would that alter the calculus in these cases?
Humans inherit mitochondria only from the mother, and an egg cells is about 0.12mm vs a sperm cell at 50μm long.
Edit: fixed a word
What controls this production?
AFAICT, only DNA and RNA fit the bill.
“Information” here is not referring to the molecule for eye color. It is referring to the information about when and where the gene is to be expressed (ie the molecule is created).
I think this has been known for a while though. That deriving molecular folding and structure from DNA sequences is a tremendously hard problem, and that “command and control” of the entire apparatus is at least as hard or harder of a problem.
There are lots of endogenous retro viruses (ERVs) in various states of genetic activity. However the only time this comes up is when considering ERVs from e.g. pigs for organ transplants. I remember reading some time ago about an effort to remove all the ERVs from a line of pigs to make that safe. I don't remember what happened of that.
You do see some unicellular organisms that will prioritise maintenance of proteins over DNA if the proteins have low copy number and are critical for survival.
Incredibly complex structure can arise from simple rules, see Turing Patterns https://en.m.wikipedia.org/wiki/Turing_pattern and cellular automata.
What does it prove though? We already know that cellular components are the same or similar across all life.