Nah they don't. Skipping over the (common on the internet) parochiality of assuming that US military practice is just 'what the military does'...
The US military certainly uses UTC/Zulu time for time stamping messages and coordinating operations globally, and timekeeping during extended missions (like submarine patrols, or air missions), but military bases and even surface vessels underway keep somewhat local time. They don't use Zulu time - that would mean the date changes in the middle of the day. You wouldn't be able to use words like 'next Tuesday' without having to disambiguate. It'd be crazy.
Zulu - Z - is just the last of the alphabetically named military timezones: https://en.m.wikipedia.org/wiki/Military_time_zone, which is a NATO standard.