What Eseutil switch should I use to verify the integrity of a database?

That depends on exactly what you want to verify. The /g switch (which the Eseutil documentation refers to as integrity check mode) scans the database header for consistency and performs some other basic checks. The /k switch (which the Eseutil documentation calls checksum mode) walks the database and computes the checksum for each page, comparing that checksum with the stored value. The latter check occurs automatically during an online backup but doesn't happen during a Microsoft Volume Shadow Copy Service (VSS) backup. To verify the quality of a VSS snapshot, use Eseutil's /k switch.