There is nothing new in the arguments of the New Atheists. They are borrowed from Feuerbach, Marx, Freud, Nietzsche, and Sartre. And what all the atheists, new and old, have in common is a mistaken notion of God, for to a person they construe God as one being among many, an item within the nexus of conditioned things. The roots of this misconception are deep and tangled, stretching back to antiquity, but I would put a good deal of the blame for the present form of the problem on the transition from an analogical to a univocal conception of being, on display in Duns Scotus and especially William of Occam. On Aquinas’s analogical interpretation, God is not one item, however impressive, in the genus of existing things. Indeed, Thomas insists that God is not an individual and is not to be categorized in any genus, even that most generic of genera, the genus of being. God is not so much ens summum (highest being) as ipsum esse subsistens. But if, as Scotus and Occam would have it, being is a univocal term, then God and creatures can be considered under the same ontological rubric, and they do indeed belong to an identical genus. This means, in consequence, that God, though he might be described as infinite, is one being among many, an individual alongside other individuals. Occam would state the principle with admirable economy of expression: Praeter illas partes absolutas nulla res est (“Outside of these absolute parts, there is nothing real”).
I realize that this might seem the very definition of medieval hairsplitting, but a great deal hinges on this point. On the analogical reading, all of finite reality participates in the fullness of the actus essendi of God, and hence God and creation cannot be construed as rivals, since they don’t compete for space, as it were, on the same ontological grid. But on the univocal reading, God and creation are competitive, and a zero-sum game does obtain. The Reformers were massively shaped by the nominalist view that came up from Occam, and they therefore inherited this competitive understanding of God’s relationship to the world, which is evident in so much of their speculation concerning justification, grace, and providence. If God is to get all of the glory, the world has to be emptied of glory; if grace is to be fully honored, nature has to be denigrated; if salvation is all God’s work, cooperation with grace has to be denied. When this notion of God became widespread in Europe after the Reformation, it provoked a powerful counter-reaction, which one can see in almost all of the major philosophical figures of early modernity. The threatening God must be explained away (as in Spinoza), fundamentally identified with human consciousness (as in Hegel), internalized as the ground of the will (as in Kant), or shunted off to the sidelines (as in most forms of Deism). In time, the God of late medieval nominalism is ushered off the stage by an impatient atheism that sees him (quite correctly) as a menace to human flourishing. Thus, Feuerbach can say, “Das Nein zu Gott ist das Ja zum Menschen,” and every atheist since has followed him. Jean-Paul Sartre, in the twentieth century, captured the exasperation with the competitive God in a syllogism: “If God exists, I cannot be free; but I am free; therefore, God does not exist.” And Christopher Hitchens has restated the Feuerbach view, observing that believing in God is like accepting permanent citizenship in a cosmic version of North Korea.
On further reflection, I was rather struck by the Latin quote from Ockham that Barron references: "praeter illas partes absolutas nulla res est". He says by it Ockham is stating a principle, but Barron does not tell us what the principle is. That God is in a genus? That being is univocal? That being is a genus? That God is one individual alongside other individuals? Does Barron think that, according to Ockham, God and creatures are parts of some whole?
The quote comes from Ockham's Ordinatio I d. 30 (OTh IV, 317). This question is a debate about relations. Scotus had held that relations were really distinct from their terms, and this question of Ockham's consists of quoting Scotus' arguments verbatim and refuting them. So it is odd, to say the least, for Barron to group Scotus with Ockham here. The quote is from some exegesis of Aristotle's Metaphysics XII. Ockham, annoyingly, in this question only talks about what can be known by pure reason, and so he does not talk about God at all, save in this reply to Scotus. So it clear why Barron would want to cite this passage, since even though there are many mentions of Ockham's view int he question, that relations are not really distinct or absolute beings, only substances are, none of them directly concern God. But Barron does misrepresent the context of the quote, I would say.
But this is not a question about univocity. And even the conclusion is not terribly controversial, i.e. that relations are not absolute things or really distinct. It was quite common in medieval thought to argue about this, long before Ockham and Scotus, and I see no connection to the univocity debate. There was some debate about whether quality and quantity were absolute categories like substance, Ockham here is denying that relations are real or absolute.
Barron's comments are then turning a rather mundane medieval position into a consequence of univocity, even though people who deny univocity could hold it and those who hold univocity could deny it. Indeed, Scotus holds univocity but is the target of Ockham's attack in the very passage Barron quotes. But there are no rules in the narrative game.
The ultimate root of Barron's comments here is the neoplatonist strand in academic thomist theologians, combined with continental speculation. The dependence of creation on God entails, for modern thomist theologians, the idea that humans cannot be granted any ontological standing in their own right, even though even on thomist natural philosophy human beings et al. are all substances, are individuals, have causal powers, support accidents, etc. And oddly, God is somehow not an individual, but way way beyond such mundane notions, though they never bother to explain that God isn't just a universal or an abstract object, what one normally would contrast an individual with.
So in the end, Barron is sort of right, at least that Thomism and Scotism are incompatible on many points. For Thomism, God is subsistent being itself, for Scotism God is an infinite This. The Scotist view for Barron entails competition between God and creatures; Barron thinks that this competition is bad and leads to protestantism. But competition would seem to be a datum of the human experience, at least if Barron's other views, are correct, that is, given that he is a Catholic Bishop he believes in human sin, and what is sin other than the assertion of one's own will and choice over the divine law/wish/choice? But there was sin prior to Scotus' theory of the univocity of the concept of being, so I think we can absolve univocity of the competition charge, at least.