Is there any reason why I can’t require this person to sit at their desk?

No, there isn't. They already indicated there is no problem with their workspace, therefore you can require them to use it. I'd also be talking to them about their performance. Both refusing to use their desk and low performance are getting into disciplinary action zone.

It's obviously not a valuable employee so no issue if they quit in a huff. And at best it will make them tell you what the actual issue is and you can move forwards with more information.

Find the root cause, you are compelled to do that

TL;DR

These are warning signals. An under-performing employee, that does not want to sit close to their colleagues, and does not even dare talk about it with you? This may be a case of workplace harassment.

You are most likely compelled to act on that... legally, contractually and morally, in order to eliminate risk to your employer, to your employee, and to yourself.

Long answer to follow...

Is this really a problem?

First you need to find out in what way is this a problem? If you disregard that this rubs you all the wrong ways, what are the downsides of this person doing this?

If you find no such downsides, then there is no problem other than that it ruffles your feathers a bit, but you can put up with that, can you not?

However, no matter if you find no such downsides or if you do find them, at least one of the following two questions need be answered.

a) What is the root cause of the behaviour?

Ask your employee again: why are they doing it that way? If they feel they do not want to answer, ask "Why do you not want to answer, is it a sensitive issue? Do you want to talk in private about it? Would you like to have a confidential representative talk to you about it and bring your wishes to us?".

The person has a reason. If you think their behaviour is a problem you need to find out if their behaviour stems from a trivial non-important reason, or if it is caused by an even bigger problem. Maybe the person has some kind of issue they are embarrassed to talk about, like a phobia for germs and one of their colleagues is being messy in a way that sets it off. Maybe there is some kind friction between them and another employee; their personal chemistry being volatile for some reason. Or maybe they are the victim of bullying, or even worse: some kind of blackmail.

It their behaviour truly is a problem, you cannot just attack the symptom (them sitting in the common area); you need to find out why this is happening, or you might very well be squeezing your employee between a rock and a hard place, or worse: failing to fulfil your duties as an employer (see more below).

Once you know the root cause, you can start working on a solution.

b) How can we work around it?

If their behaviour truly is a problem, and the root cause for this cannot be found or it is of no interest to you as long as they perform well, try to find a solution around this problem. Can they work in another part of the building? Can they telecommute? Would they consider another assignment? Tell them that this is a problem for the employer, and that a solution must be found... and tell them that you welcome hearing solutions from them.

Why not just make them go back to their place or kick them out?

Because by US federal law, employers have a duty to act against discrimination, bullying and harassment.

The employer is automatically liable for harassment by a supervisor that results in a negative employment action such as termination, failure to promote or hire, and loss of wages. If the supervisor's harassment results in a hostile work environment, the employer can avoid liability only if it can prove that: 1) it reasonably tried to prevent and promptly correct the harassing behavior; and 2) the employee unreasonably failed to take advantage of any preventive or corrective opportunities provided by the employer.

The employer will be liable for harassment by non-supervisory employees or non-employees over whom it has control (e.g., independent contractors or customers on the premises), if it knew, or should have known about the harassment and failed to take prompt and appropriate corrective action.

There is obviously something unusual going on with this employee. It may be that they are just being eccentric. But if they are not, and this is indeed a symptom of a bigger problem — such as workplace harassment, bullying or discrimination — the employer has a duty to act.

If the employer fails to act, and this comes back to haunt them in the form of a civil suit, they will be asking around. They will be asking "Did anyone notice anything out of the ordinary with this person?". Well you obviously did; you noticed something very out of the ordinary; you noticed something so much out of the ordinary that you went on The Workplace SE to ask about it.

The question itself is now evidence that you noticed something was off with this employee.

When the court then asks the employer "Why did you fail to act on this signal?", I guarantee you that the answer "Well... anonymous people on The Workplace Stack Exchange said we did not need to but could instead just force the employee to go back to their place" will not suffice as an answer.

This means your employer will be hung out to dry by the court. Which in turn means they will be looking for whoever are responsible for putting them in that fix. And that would be you, because I do not think I am going very far out on a limb here when I say that it is most likely a part of your job description, and of your workplace employee policies, to be on the lookout for warning signs of harassment and other things that your employer is legally required to work to prevent.

Summary: yes, there are reasons

You ask...

Is there any reason why I can’t require this person to sit at their desk?

Yes, there are such reasons, in that you have three very strong warning signals going off here: 1) the worker is under-performing 2) they do not want to be around colleagues 3) they do not dare talk to you about it. Something may be wrong here, and now that you have picked up on this unusual behaviour, you are then duty-bound to act.

Most likely this is a symptom of something. You need to find out what that something is, or at least find a way to work around it. It may be innocuous, but it may also be a symptom of a problem that your employer is legally required to deal with. This in turn means that your employer expects you to be on the lookout for such things and bring it up if you suspect it might be happening.

Hence, simply nagging or forcing your employee to comply without seeking to know why they do what they do, is setting yourself up for a bad ending of this story, for the employee, for your employer, and ultimately: for you.

First, if you are a manager-type person you want to get HR involved. If you aren't a formal manager -- for example, a "team lead" or "group supervisor" -- you should get someone who's formally a manager involved.

Assuming that you can justify the claimed lack of productivity or performance through some set of metrics, and the person has refused to correct their behavior, you should have a valid reason to separate ("fire") the employee -- assuming there is nothing going on otherwise. Insubordination is usually a valid cause for separation.

The reason I strongly suggest you get Human Resources involved is because employees can have issues which they do not want to share with their manager. Bullying and subtle forms of harassment come to mind, along with cultural differences which are creating friction. I've had "how to be a manager" courses in the past and "my co-worker smells bad / talks loud / make off-color jokes of non-protected classes / etc." are common topics. If the co-worker who's creating the issue is a well-established or favored employee, going to the manager with the complaint can be perceived as career limiting.

It is important to keep in mind that seemingly silly reasons for not wanting to sit in a specific location can be very real. At one employer the lighting was so bright it was seriously impacting our performance, so we removed tube lights to make our area more hospitable, but some people on the team liked the bright lights, so they wanted to sit where there was more light. At another job, my position required that I interact with a lot of employees from other departments and my office mate asked to be moved to another office -- in that case, I was moved to my own office so I could have side chairs for visitors when they came.

What's most important is that you dial-down the strong-arm techniques and as another responded said try to find the cause of this behavior. If after getting HR involved there is still no resolution you have to decide if they really are causing a problem and not simply rubbing you the wrong way. Once you have all those answers you should have either the information needed to correct the problem (for example, move to another location with better lighting, away from an A/C vent, away from a "busy" co-worker) or the documentation needed to separate the employee.

Best of luck.

Yes, you can require that, but it seems odd that your rapport with the employee is such that you have no clue. How hard have you tried. Is there a good climate at your work place (not just from your position) ? Is the employee a junior / newer employee? Or the employee a female that comes from a culture where they aren't expected to speak up?

The root cause could be something as simple as an embarrassing issue; maybe one of your other employees has a body odor or flatulence problem and your employee in question doesn't know how to deal with it without an awkward result. This is not uncommon. I dealt with it as the employee, and as the boss of a group of people.

At my first job in Atlanta, when I was young out of school, one of our senior developers constantly passed terrible gas. Nobody wanted to discuss it. Took me weeks to figure out who it was once I started the job. It started to make me nauseous. Other folks would walk through and associate the smell with our area in general, and that also bugged me.

After my 6 months initial contract was up, I left the job. I got a better offer, but I am not kidding, I was young and new and didn't want to deal with the awkwardness of reporting the employee, and I preferred to find a new job. I was happy to leave that baggage behind.