For the most part, no. When it came to the Army, placement of enlistees and draftees after they had signed up or were drafted (and as a result, eventual theater assignments) was generally completely random, pursuant to the calculated needs of the service at that time. All-volunteer Army units such as the Rangers or parachute infantry units filled their quotas by selecting men who had placed a preference after enlistment or induction, and then met the requisite physical and mental qualifications. For the experience of the Army draftee, see this earlier answer I gave. The same kind of system existed for the Navy and the Marine Corps, except that they did not begin tapping the pool of draftees until early 1943.

There were six draft registrations during WWII:

Draft Registrations:

Registration	Date	Criteria for Registration
1	October 16, 1940	Men who were 21-36 years old
2	July 1, 1941	Men who had turned 21 since October 16, 1940
3	February 14-16, 1942	Men who had turned 20 on or before December 31, 1941, and were not older than 45 by February 16, 1942
4	April 25-27, 1942	Men who were at least 45 on or before February 16, 1942 and not older than 65 on or before April 27, 1942
5	June 30, 1942	Men born after January 1, 1922 and before June 30, 1924
6	December 10-31, 1942	All men who had turned 18 by December 31, 1942
Extra	November 16-December 31, 1943	Men living abroad aged 18-45

After the sixth registration, all men who had reached their 18th birthday were required by law to register with their local draft boards.

With the signing of Executive Order 9279 on December 5, 1942, President Franklin D. Roosevelt eliminated voluntary enlistment into the military for those men who were between 18 and 37 years old. Those men who were 17, and those who were 38 to 45 (i.e. those men who were considered old enough for service, but not actually subject to the draft) could still enlist voluntarily into the military provided that they met its qualifications for service.

After the effective date of this Order no made person who has attained the eighteenth anniversary and has not attained the thirty-eighth anniversary of the day of his birth shall be inducted into the enlisted personnel of the armed forces (including reserve components), except, under provisions of the Selective Training and Service Act of 1940, as amended; but any such person who has, on or before the effective date of this Order, submitted a bona fide application for voluntary enlistment may be enlisted within ten days after said date.

This order was made so as to not disrupt vital war industries and farm labor by having men essential in these occupations run off and join the military. It also controlled how many men the military received at a time. The Selective Service set a quota of drafting around 200,000 men per month.

In contrast to the Army, the Navy, until the signing of Executive Order 9279, remained "free" of any Selective Service constraints due to sufficient enlistments and a rocky relationship with the Service itself. The Navy had refused to participate in the Selective Training and Service Act of 1940 (a request granted by the President himself) and had complained several times to the Secretary of War as to why it should be able to remain volunteer-only. Extremely aggressive recruiting processes (including assigning officers with sales backgrounds to recruiting offices and an extensive advertising campaign) had annoyed the Selective Service to the point where, in early 1942, they refused to provide the Navy with lists of men from local draft boards classified 1-A!. Voluntary enlistments, especially after Pearl Harbor, managed to carry the Navy to Executive Order 9279.

The Marine Corps (as a part of the Navy) also used an entirely voluntary system of enlistment through the first years of the war, and this proved to be successful. With the war on, it was realized that utilizing the draft's manpower pool would be necessary to provide enough Marines. The Marine Corps issued their first personnel quota on February 1, 1943, with the first draftees arriving to training later that month. Two weeks before President Roosevelt signed Executive Order 9279, the Marine Corps had arranged to send liaison officers to state and local draft boards to identify inductees who had made an explicit request to serve in the Marine Corps upon draft registration, or those that wished to. The liaison officers had the authority to funnel willing men into the Marine Corps or postpone the inductions of those men who wished to join the Marine Corps until vacancies existed (see below) They also had the authority to discharge prospective Marines from the Selective Service System, to allow them to re-enlist voluntarily and avoid the stigma among Marines of being drafted. Of 224,323 Marine draftees, less than 70,000 chose to keep their "inductee" status. Of the 275,985 men who entered the Marine Corps between February 1, 1943 and July 31, 1945, 58,927 were 17 year old volunteers.

Each month, the services submitted a manpower requirement to their respective Secretary, who then presented this figure to the Director of the Selective Service. At Selective Service Headquarters, a total call was made up. Quotas of men needed were then issued to the states, where they were divided up among the local draft boards. A ratio was calculated between the Army and Navy (which included the Coast Guard and Marine Corps)

Each of the induction stations allotted inductees on the basis of this ratio. To assure equitable distribution of manpower by quality, categories were set up according to age, education, and occupational skill. The quotas for the Army and Navy were to be made up of proportionate numbers from all these categories. The Navy quota at each induction station was then broken down into Navy, Marine Corps, and Coast Guard components, again by use of a ratio. Selectees at induction stations also had the option of picking their branch of service, provided vacancies existed in the quota of the service of their choice.

Sources:

Executive Order 9279

Administration of the Navy Department in World War II, chapter VII, by Rear Admiral Julius Augustus Furer, USN

Administration of the Navy Department in World War II, chapter XIV, by Rear Admiral Julius Augustus Furer, USN