Dynamical few-body encounters in the dense cores of young massive star clusters are responsible for the loss of a significant fraction of their massive stellar content. Some of the escaping (runaway) stars move through the ambient medium supersonically and can be revealed via detection of their bow shocks (visible in the infrared, optical or radio). In this paper, which is the second of a series of papers devoted to the search for OB stars running away from young (several Myr) Galactic clusters and OB associations, we present the results of the search for bow shocks around the star-forming region NGC 6357. Using the archival data of the Midcourse Space Experiment (MSX) satellite and the Spitzer Space Telescope, and the preliminary data release of the Wide-Field Infrared Survey Explorer (WISE), we discovered seven bow shocks, whose geometry is consistent with the possibility that they are generated by stars expelled from the young star clusters, Pismis 24 and AH03 J1725-34.4, associated with NGC 6357. Two of the seven bow shocks are driven by the already known O stars. Follow-up spectroscopy of three other bow shock-producing stars showed that they are O-type stars as well, while the 2MASS photometry of the remaining two stars suggests that they could be B0 V stars, provided that both are located at the same distance as NGC 6357. Detection of numerous massive stars ejected from the very young clusters is consistent with the theoretical expectation that star clusters can effectively lose massive stars at the very beginning of their dynamical evolution and lends strong support to the idea that probably all field OB stars have been dynamically ejected from their birth clusters. A by-product of our search for bow shocks around NGC 6357 is the detection of three circular shells typical of luminous blue variable and late WN-type Wolf-Rayet stars.