No. Grocery store jars should not be reused for home canning because they're usually not as thick as home canning jars (making them more likely to shatter or explode when heated), and their necks are often too short to allow for proper sealing.
The good news is that canning jars are fairly inexpensive and can be reused for years and years.
A Word of Caution: Don't be fooled by grocery store jars that come inscribed with the word "Mason." It's usually there to give the product a "down home" feel, and should not be considered an indicator of the jar's quality.