Personally I don't like carpet tiles because most of the time they don't hold up well. That is usually due to an improper installation. With that said if you do your prep work and installation properly you probably won't have any problem. Never the less I would not put it in my home, but that is just me.
Carpet tiles are ok, as long as they only sit on the cement and are not glued down. Primary reason is cleaning due to microbial development, dust mites and odors. Carpet by itself can become pretty nasty. When we do carpet checks in basements, picture a mini Godzilla movie under the microscope. You would be amazed at what develops on damp cement floors.
Floors not damp? Do not bet on that. All cement floors have moisture issues, unless you are 100% sure that the vapor barrier that is supposed to be placed down prior to the cement pour was properly done with no punctures and all seams were taped and not just overlapped.
You have radon? Then the vapor barrier has failed. As the plastic will keep the radon gas from getting into the living area.
Ideally tile floors or simply painted or stained floors is best. Does not harbor mold and is easy to clean. Carpet runners and large throw rugs can be used for comfort. Carpet tiles also can be used as long as you remove them and clean both sides.
Forget about using steam or chemical cleaners from the guys in the white vans that clean carpets, Mold loves steam. So does the dust mites that thrive in moisture.
Have children? Earaches? Can be caused by dust mites. Remember children breath in a different air zones then adults. They spend their time laying on floor and rolling around. Adults, well most adults sit up and are not exposed to the same air that their kids are.
So if your able to remove and properly clean. You will be better off. But if the carpet squares are fastened down with any sort of cement. I would suggest against this practice.
I'm wary about any carpet in the basement. Too much risk of moisture getting trapped in that carpet. Tile or finished concrete is much easier to clean up and dry out without any further mold risk.
You might want to check out Flor carpet tiles, Sherry. There's no glue involved. They are held together with sticky discs. And, if any wear out, get stained, etc., the company will send you a pre-paid return label to send them back to their Georgia plant for recycling.
I'm with Steve on the carpet in the basement option. Don't know how elaborate the decor is but if soft under-foot is the object take a look atr some of the new interlocking foam floor tile products. Comfortable, easy in and out and less subject to moisture generated problems.
Wouldn't be putting carpet tiles in a basement....as the others have suggested, try to stay away from carpet if you can and install ceramic tile, engineered flooring that has been sealed on the backside or decorative concrete. You'll find some great vendors here who can do any of these.
I'm with the folks above. Carpet and basements don't mix. There is always a certain amount of moisture that can get in the carpet, then you have mold problems, which if bad enough, can cause respritory issues Also if folks have allergies it can be bad news