Debris disk stars are good targets for high-contrast imaging searches for planetary systems, since debris disks have been shown to have a tentative correlation with giant planets. We selected 20 stars identified as debris disk hosts by the WISEmission, with particularly high levels of warm dust. We observed these with the VLT/SPHEREhighcontrast imaging instrument with the goal of finding planets and imaging the disks in scattered light. Our survey reaches a median 5σsensitivity of 10.4 M J at 25 au and 5.9 M J at 100 au. We identified three new stellar companions (HD 18378B, HD 19257B, and HD 133778B): two are mid-M-type stars and one is a late-K or early-M star. Three additional stars have very widely separated stellar companions (all at >2000 au) identified in the Gaia catalog. The stars hosting the three SPHERE-identified companions are all older (700 Myr), with one having recently left the main sequence and one a giant star. We infer that the high volumes of dust observed around these stars has been caused by a recent collision between the planets and planetesimal belts in the system, although for the most evolved star, mass loss could also be responsible for the infrared excess. Future mid-infrared spectroscopy or polarimetric imaging may allow the positions and spatial extent of these dust belts to be constrained, thereby providing evidence as to the true cause of the elevated levels of dust around these old systems. None of the disks in this survey is resolved in scattered light. Unified Astronomy Thesaurus concepts: Exoplanets (498); Extrasolar gas giants (509); Debris disks (363); Circumstellar disks (235); Planetesimals (1259); Direct imaging (387); Coronagraphic imaging (313) Supporting material: data behind figure, machine-readable table