There's two features in Marxism (here I'm working from its Hegelian background primarily as I think these problems transfer) that conspire to enable the objection that it is anti-individual:
First, following Hegel, the Marxist picture is such that the whole is the real. In the Hegelian picture, this whole is "spiritual" but in the marxist version it's somehow merely a material whole. Consequently, the individual is not of prime importance as in views like existentialism. Instead, the individual is a part of something larger.
This by itself does not eliminate the place for the individual. But it conspires with a further feature to do so. On this sort of picture, there's a sense of how the individual should then relate to and be a part of said social whole. One specific place this comes up is in the idea of what thinking is to be used for.
On the Hegelian version, the way in which the individual should think is necessitated by the structure of thinking. Hegel specifically excludes revelations and geniuses from his idea of how we should think. In other words, every being gets a part to play in the inevitable unfolding of "progress" in the Concept whether as a thinking bit or a material bit. This happens dialectically so that each element is to be accepted in all of its difference. The rub being that on this picture you cannot choose not to advance.
Thus, for instance, if as Hegel claims Christianity is superior to Judaism, then making the individual choice to follow Jewish religion (all his terms here -- not me), then you're just plain wrong. Similarly, if your form of Christianity involves personal faith, it's also wrong for Hegel. In other words, progress, freedom, and thinking, but all have a predetermined course. (See Charles Taylor, Hegel (Cambridge 1977), p. 185 and Phenomenology of Spirit 415)
My sense is that Marx gets rid of the thinking bits (as a Hegel scholar I don't really know how this part works) but winds up with a similarly deterministic (and thus anti-individual) account of thought and progress. For instance, you're not allowed to decide that you don't mind being alienated from your labor; that just has the role of sin in previous views of being something you are bound to be if you don't properly relate to your work.
For Hegel, this would not be anti-individual or anti-freedom, but this reply to the objection hinges on some fancy footwork. If Hegel is right about the nature of consciousness and thought and then metaphysics, then it follows that freedom is the self using reason to pursue the goals of reason. And the goals of reason are deterministically knowable. In other words, Hegel is pro-individual and pro-freedom if it is the case that the Concept determines what the individual should do and has them pursue it.
But this is a very unsatisfying reply if you're committed to a more robust concept of individual or freedom. Or to put it another way, if you believe in a freedom as incompatibilism such that actions are free when the individual can choose their actions or their action-shaping preferences without regard for a unified idea of progress, then this is unconvincing.
To make it a bit more practical, for Hegel, you should contribute to society, and society will have certain ideals and values. To pick a near contemporary example, for a while the laws regarding gay marriage varied by state. If it was necessary that gay marriage be allowed, then states restricting it could not be allowed. Since it's become legal in all states, we've also seen periodic clashes that take it further -- should individuals be required to acknowledge or participate in the gay marriages of others? If there's a unified idea of progress and something is part of it, then on the Hegelian picture, there's no right of conscience to refuse to accept it or participate in it.
Hegel doesn't find this problematic precisely because society trumps the isolated individual but incorporates the individual qua living, reasoning bit of the whole. If what we want is individuals who are free to be separate, then Hegel opposes this as an illusion (an immediacy and unmediated state).