I read that Hegel critiques Kant's assumption of the universal concepts, that all humans share the same concepts exactly. But at its core, isn't this statement(claiming an overarching belief of man) true?
The answer to this question is complicated by recent Kant scholarship.
Basic CPR Kant
The picture I see of Kant in the Critique of Pure Reason is that in the "Transcendental Deduction of the Pure Categories of the Understanding", Kant identifies exactly 12 categories (no more and no less and no different than his list) that he thinks is true of the understanding (A70/B95-A93/B109) (See http://plato.stanford.edu/entries/categories/ ). In addition to this, Kant believes that we form representations under the forms of sensibility (space and time).
Thus, Kant gives us a picture where our cognition takes "things" and renders them sensible under space/time as "representations" and then turns them into "objects" as the 12 categories are applied.
If that's what you mean by "concepts" in your question, then Kant does think we share these things. Whether they are innate is not very important to Kant. What matters is that these are the structures of pure reason.
Hegel agrees with Kant that we bring "categories" to what we think about. For him, the key is that we bring Begriff (often translated as "concept"). But for Hegel, it's not a list of 12 things we bring but rather an evolving apparatus of understanding ourselves and our world (short version as the task of Phenomenology of Spirit: we move from trying to understand and locate truth in objects outside of ourselves to focusing on ourselves as the sort of beings that bring truth to things and in the process we also move from "I" to "we").
One complication that might confuse the contemporary reader is that neither (is at least in his own terms) making a claim that the human brain brings with it certain organizing apparatuses as the system in which by necessity operate. Instead, both are raising a thesis about mind and reason. They are claiming that reason does these things with little respect for biology. Now, you can rearticulate similar ideas with modern neuroscience (though if you want to share Kant's commitment to 12 categories exactly you'd be in lonely company).
A lot of contemporary Kant scholarship (especially in Kant's moral philosophy) is not committed to the precise articulation Kant gives in the CPR. Instead, they are interested in a looser commitment to reason. Thus, for instance, Christine Korsgaard, gives her own deduction for how we find ourselves committed to rational action with no little if any reference to the categories of the understanding.
Other Kantians, such as Jurgen Habermas, openly admit that they accept Hegel's critique of Kant's static view of reason and the faculties, but they find Kant's picture for moral philosophy attractive.
Similarly, texts like the Cambridge Companion to Kant will include chapters like Beatrice Longuenesse's chapter on the categories Kant has, but these are meant to explain Kant's view rather than defend the truth of the claim there are 12 categories.
Whether the overarching claim that humans bring concepts to what they do is true wouldn't eviscerate Hegel's critique, because (a) Hegel agrees we bring concepts to things as objects and (b) Hegel's critique is about the way that Kant thinks that works, and (c) contemporary Kant scholars might not accept the nitty-gritty of Hegel's critique but they don't defend Kant's conception of what we bring.