Yes. You can pull up your existing carpeting and replace it with carpet tiles.
That may make it easier to diagnose issues with the floor/sub-floor without having to pull up existing carpet and remove the furniture from a room, but it will do nothing to actually fix the problem.
It's a guarantee that you'll have to pull up the existing carpet and more furniture in order to replace your current carpeting and you'll still be faced with the initial problem.
On the other hand, carpet tiles don't necessarily come up easily, especially in high-traffic areas where squeaking floors are most common.
Once you get a carpet tile up, putting it back down is another matter entirely. Sometimes they are damaged to the point that they need to be replaced. Sometimes not. Sometimes they just never look right afterward. The success of replacing the tile depends on the effects of aging exhibited by the original carpet and, to some extent, the availability of a replacement tile.
Another thing to consider is that squeaking floors can usually be fixed without having to remove any carpeting (or furniture). There are many questions and answers here (and the rest of the Internet) that deal with fixing a squeaky floor.
If you need advice on how to do that, please ask another question here as it seems to be outside the scope/intent of your original question.
