The real answer is "it depends". Military forces are designed to be adaptable and come up with strategies and tactics against an intelligent opponent (there is an old military saying "the enemy gets a vote, too").
Anon has set the stage with a good answer, and there will indeed be a time where it is unclear what is going on and units may be overwhelmed by zombie infections from within or without. However, at some point the incredible truth finally becomes clear, and surviving units will be able to come up with some plans for action.
At its base, a zombie plague can and should be treated like any other outbreak of disease. Quarantine of infected people and areas, strict control of movement and delivery of medication if some form of anti-zombie vaccine is possible. The military members also can take heart in the fact that a virulent plague like the zombie outbreak will burn itself out in fairly short order. With areas quarantined and movement restricted, zombies will rapidly run out of new people to infect, and biologically speaking, the infected bodies will run out of energy once everyone who can be eaten is consumed, and themselves die.
Once an area has gone "cold", specialists can carefully enter and examine the area, and scouts and snipers move in to clear any remaining zombies. The dead will likely be burned to prevent pockets of disease breaking out in the rear area. The perimeter expands and is secured, and the cycle begins anew.
In many ways this also resembles the classic anti-insurgent tache d'huile strategy, but can also be considered a variation of the British "blockhouse" strategy used during the Boer War (tache d'huile dates back to the late 1800's, and the blockhouse strategy was used in the early 1900's, so it isn't like you have to invent anything new).
The greatest difficulty for the US military is the "force to space" ratio; even if every Active, Reserve and National Guard member is unaffected and working on the problem, there are far too few service members to effectively cover the Continental United States. What will happen instead is a drawn out campaign where soldiers using low tech gear like razor wire fencing and 12 gauge shotguns enclose areas to secure them, then wait until the next district is "cold", rinse and repeat. After a few months, the vast majority of the zombies will no longer be active, and the remaining areas of the United States not already quarantined will need to be systematically covered and disinfected. Once again, this will be slow and heavy work, especially if the logistical chain has been disrupted, but still doable.