A good rundown of arguments against a universal grammar is given in this article (There is no language instinct) at Aeon. Some of it is a little sloppy, especially a line on genes near the end. Though I have only tepidly read on such stuff, the Chomsky line, the innate language faculty, as it is often presented, is surely wrong. Jesse Prinz makes similar arguments, as those in the article, in his book Beyond Human Nature. Also, Patricia Churchland gives an enjoyable rebuttal in Brain-Wise:
Nevertheless, it is sobering to remind ourselves of the many cognitive artifacts that are known to be cultural inventions. . . We do not know how much of the complexity seen in human language depends on cultural evolution. Structural similarities among human languages are consistent with, but certainly do not entail, that there exists a genetically specified grammar module in the human brain. Such structural commonalities as do exist could be as well explained, so far as is known, as arising from similarities in fundamental aspects of human experience, such as spatiality, sociability, the need for sequence assembling in forming plans and in behavioral execution, and so forth. As Elizabeth Bates wryly commented, the similarities among humans in getting food to the mouth by using hands rather than feet does not imply the existence of 'hands for feeding module. Rather the existence of a shared body plan and the ease of hand feeding relative to foot feeding suffice to explain 'feeding universals.' (284)
I've often wondered what exactly a noun is, whether we could see it as continuous with animal communication, whether nouning and verbing is really that different, and whether it would be possible to create creatures that use complex language without using nouns. I would think you would have to do some significant jingling to get complex creatures with an ability to use language to not dialectically turn to the use of nouns. But similar to the Churchland quote, humans' use of nouns does not seem like something that would need specific structures for it. Stanislas Dehaene (see below) gives some detailed accounts of what exactly is structured for in the brain as regards reading, which he suggests is not brain structures that were selected for (against) specifically for reading capacities. If such a story is at least partially true, then the divide between what is actually genetically structured as regards the brain and what actually comes out as regards social institutions and behaviors, even if fairly widespread, is a more complicated picture than how universal grammar is often presented
Other things of interest:
Speaking of Brain-Wise (2002) by Churchland, it is recommended over her earlier book Neurophilosophy (1989). These books seemed to cover much of the same stuff, especially as regards philosophical positions, with Brain-Wise being more readable and more up-to-date.
Stanislas Dehaene's Reading in the Brain and Consciousness and the Brain (2014) are both good. I think Reading in the Brain (2010) could provide ammunition against the Chomsky line as well, which is why I was thinking about it. I say that despite the fact that Dehaene may endorse the Chomsky line on universal grammar, or something of the sort. His general take on consciousness is moving in the right direction as well.