We present a deep Chandra X-ray Observatory study of the peculiar binary radio millisecond pulsar PSR J1740--5340 and candidate millisecond pulsars (MSPs) in the globular cluster NGC 6397. The X-rays from PSR J1740--5340 appear to be non-thermal and exhibit variability at the binary period. These properties suggest the presence of a relativistic intrabinary shock formed due to interaction of a relativistic rotation-powered pulsar wind and outflow from the unusual "red-straggler/sub-subgiant" companion. We find the X-ray source U18 to show similar X-ray and optical properties to those of PSR J1740--5340, making it a strong MSP candidate. It exhibits variability on timescales from hours to years, also consistent with an intrabinary shock origin of its X-ray emission. The unprecedented depth of the X-ray data allows us to conduct a complete census of MSPs in NGC 6397. Based on the properties of the present sample of X-ray--detected MSPs in the Galaxy we find that NGC 6397 probably hosts no more than 6 MSPs. ; Comment: 10 pages, 6 figures, 3 tables; accepted for publication in The Astrophysical Journal