Standard buttons on button up shirts (http://jeevesny NULL.com/services/menswear-cleaning-and-restorations-nyc/mens-shirt-cleaning) should not have any issues when entering the dry cleaning processes (http://jeevesny NULL.com/services/dry-cleaning-new-york-city). But those that are made of natural materials, for example, shell and bone, may chip, crack or break. Non luxury dry-cleaners often do not protect these buttons by removing them before dry cleaning, wrapping them in aluminum foil or wrapping them in special velcro covers. If your buttons are breaking on a newer item, this is probably why.
At Jeeves (https://jeevesny NULL.com/), we remove any button prior to cleaning that we feel is irreplaceable or may be damaged during our cleaning process. And we actually even employ 2 dedicated staff members whose only job is is to remove buttons from garments before cleaning and re-attach them after cleaning. This is the only way to ensure that the entire look of the garment stays true to the way in which the designer intended.