Would it really be too far to argue that corporations should not have rights? More specifically, I would like to argue that corporate persons do not have rights on par with natural persons. I don't think this is really all that implausible. First and foremost, let's think about why natural persons have rights (when they do). The general trend among democratic theorists is to argue that rights claims among democratic citizens arises from something like mutual consent and recognition of one another as autonomous agents. I'm an autonomous agent, and you're an autonomous agent. If we're going to maintain that autonomy and cooperate, we should guarantee one another that we'll respect one another's autonomy. A system of rights is established that guarantees that we can each do as we please as long as we don't interfere with anyone else doing the same.
It's a nice fairytale, and it can be told through the lens of game theory, mutually uninterested contractors, or deliberative rational agency. Kant tells something like this story, as does Mill. Rawls draws his version from both of them. Habermas also tells it. Honestly, despite my usual skepticism of often-told tales, I like this one. Even if it is not a fully accurate description, it sets out a norm that democratic societies should adopt. Democratic citizens should recognize themselves as free, autonomous agents consenting to live among other agents through mutual recognition of that status.
Corporate entities are a different story. They are created as legal fictions to serve the ends of citizens. Insofar as a corporate entity is recognized as a person, it is to allow it to own property, establish credit, and absorb risk for the sake of its human managers, investors, and employees. At the end of the day, a corporate entity is made to serve a person, not the other way around. We do not recognize corporations as autonomous because they are not autonomous in either a strong or weak sense. Corporations do not make decisions. People make decisions and execute them through the corporation. Furthermore, corporate entities are incapable of recognizing their own autonomy or the autonomy of other agents. Corporate entities do not have the capacity for reflection or self-development that natural persons do. As such, they are unsuitable vehicles for rights, entirely unlike natural persons in the relevant capacities.
Nevertheless, we recognize that corporations can have property rights, can establish and maintain formal relationships (ie, client-vendor accounts), and, most crucially, operate businesses. These rights arise from a formal granting by natural persons. The legal fiction is established and recognized because the institution of corporate entities hold advantages for citizens. In other words, corporate rights are parasitic on the rights of natural persons. The rights of natural persons arise from mutual recognition as autonomous agents. One set of rights claims is prior to other, so they should not be treated as on par.
The result is a simple maxim: when considering the rights of natural persons, corporate entities must be treated as having no rights. We can say that natural persons who are stakeholders in the corporation have rights, and they likely also have attendant interests that should be respected. Nevertheless, the corporate entity itself has no rights claims with any value when the rights of a natural person are at stake. We might think of this as the price corporations pay for being allowed to absorb risk for their stakeholders.