I sometimes look at articles still to see what my parents are learning and how that frames their view of me (former member, atheist, computational evolutionary researcher).
Anyway for the proof using propositional logic:
p -> q (if we have p, then also have q)
p = information
q = attributed to intelligence and not mindless processes.
So the proposition is if there is information in something then it must be attributed to intelligence.
We can provide proof that this proposition is false. To do so all we need to do is provide a counter example where we have p, but do not have q. In fact I'll provide three for overkill.
1. Snowflakes. The exact structure of each snowflake contains information, information about the environment in which it forms. In fact this information can be said to have come about by "mindless processes," in that it is a form of order that was achieved by stochastic mechanisms. This example shows no intelligent agent was needed to encode a snowflake
2. Tree rings. Each ring of a tree contains information not just about the age of the tree, but also its growth each year (and probably other things). There is no need to attribute this information to some intelligent agent, as the mindless processes of the growth of a tree are sufficient to generate the information.
3. Ocean Floor. The layers sand on the ocean floor contains information about the environment of the sea over multiple years. There was not intelligence to the creation of this information. Rather it was generated by once again the stochastic mechanisms that are at play in ocean.
So here are three examples of "information" where people don't attribute to God or aliens. Clearly, then the Awake article is making a generalization where such is not warranted. The logic is clearly incorrect.
Of course there is the other problem of the examples they provide being weak analogies (comparing man-made, not living things to DNA). There is also the problem of incorrectly applying information theory (no sender/receiver, no predetermined alphabet, etc).