Is a cardboard box allowed as carry-on luggage?

Years ago I did this when I moved to a new country. I wanted to bring my desktop computer, so I measured it to confirm it was within the allowed measurements, and traveled with it wrapped in very soft cardboard.

I did complete my three flights without issues, but it did raise some eyebrows with the employees: at one check-in counter the employee claimed that it was not the right size, but I solved that by showing her that the computer fit perfectly in the measuring basket they had.

I've seen them used, by my family and others, and it's generally not been a problem.

A few points to consider - they're more easily used as (disposable) check in bags, because someone else will be carrying them - for a carry-on, you might want to devise a strap or the like to lift with, as it does get annoying to carry a box (particularly a heavy one, to fit the 'extra kilos') all the way through the airport, since there's fewer ways to carry it securely.

You might consider one of those super-cheap lightweight collapsible duffles as an alternative - we used to have a couple tucked into the outer pocket of the suitcases so if we ended up with something bulky or heavy, we could produce an extra 'bag' and repack on the spot to make everything fit the luggage limits.

Or take Moyli's suggestion, to use a bag to more conveniently carry the box. it's just about what's easier for you.

Be prepared to have the box X-ray'd and to open the box for inspection, other than that, I don't see why not.

You may or may not have a problem. There are no US laws or regulations about using a box as carry on luggage. TSA shouldn't have a problem* and everything should be fine, as far as rules go.

That being said you have two potential issues that you need to watch for. The airline may have rules against it. Not laws or regulations, but just rules. Most airlines follow the same basic rules, so that travelers don't get confused, but there are some differences. Specially on budget airlines (where they may charge per carry on) or "high end" airlines where they want to discourage larger carry ons. So make sure to check with your airline directly.

*The second problem is going to be "it's unusual". Around holidays and throughout the year people carry boxes with gifts quite a bit, so this shouldn't be a problem, but every time you do something different then the norm, be prepared for someone else to have a problem with it. TSA people might take an extra through look, for example, or have to call their manager or some such. Make sure to leave some extra time for that non-sense.

So to be clear, you should not have a problem at the airport, the airline may have a rule against it, but that would be unusual and it's best to call them to find out.