In Band of Brothers, a soilder is depicted bursting into tears at the news of FDR passing away, what was the overall reaction of the average soilder and how did it impact morale?

Reactions varied from indifferent to very emotional. You must remember that the United States was not as "connected" as today (i.e., via the Internet and mobile phones), and many soldiers came from overwhelmingly rural backgrounds. The majority of farms did not yet have electricity, and these men would have had nothing to communicate with the outside world besides the mail (often obtained via General Delivery, R. F. D., or a Star Route), an occasional newspaper, potentially a radio, and the family car, if they had one. The only real way these men, as well as others, would have been able to interact with Roosevelt or discuss politics would have been what they had heard over the radio, from the paper, or from people in the nearest town, if they could spare any time away from their farm work; men in cities or nearer to Washington had more opportunities to discuss politics. Many men who followed politics only knew only Roosevelt for essentially their entire cognizant lives, as he had served over three terms (12 years, 1 month, and 8 days), and were greatly impacted by his death. Also similar to today, many men were just not that politically inclined, and did not follow him much or think much of his death. An article was published in Yank, The Army Weekly, in May 1945 which had several short interviews with soldiers around the world about how they felt when hearing of President Roosevelt's death:

Before the news came over the radio from San Francisco, the GIs in the Eighth Army casual camp in the Philippines were talking mostly about the new adjusted-service-rating cards that two men, fresh from the States, had brought along with their service records and Form 20s. Then an infantryman back from morning chow said that the President had died from a heart attack. Another guy was positive that he had been killed in a plane accident on his way to Berlin to sign the peace treaty.

When they go the story straight and realized that it wasn't just another rumor, everybody in the camp was stunned and bewildered.

Pvt. Howard McWatters of Nevada City, Calif., just released from the hospital and waiting to go back to the Americal Division, shook his head slowly. "Roosevelt made a lot of mistakes," he said. "But I think he did the best he could, and when he made mistakes he usually admitted it. Nobody could compare with him as President."

Pfc. David Smith of Council Bluffs, Iowa, a railroad man in civilian life and a tank gunner in the Army, said, "Now I suppose the Germans and Japs will think they'll get something weaker than unconditional surrender. I suppose they identified Roosevelt with our country so long that they think Roosevelt is America."

There was more talk about the dead President and the new one. Somebody remarked that when you were in a casual camp in the Philippines, you were pretty far away from things. Then gradually the conversation swung back toward the adjusted-service-rating cards, and an argument started about the demobilization points.

In Rome the Allied Command closed its places of amusement and the Italian officials shut down the civilian movies, the schools, the banks and the opera. "I came out of my tent this morning," said Pfc. Fred Carlson of New York City, and the 1st Replacement Depot, "and I saw the flag at half mast. I asked who was dead. Then they told me. I hope it won't work out like when we lost Wilson after the last war."

Pvt. A. J. Smith of Naperville, Ill., an MP in Rome, was in the President's motorcycle escort when he visited Oran in 1943. "I stood very close when he came down from the gangplank from the Iowa," he said. "He looked tired and aged then."

A Navy lifeguard spread the news among the GIs and sailors on Waikiki Beach in Honolulu. Most of them walked into the extensive Outrigger Club, which is ordinarily reserved for members only, and silently sat by the radio in their swim suits, listening to the reports of what had happened in Warm Springs.

At Payne Field, the big ATC base near Cairo, Cpl. George Patcheck of Chicago was reading the story in the Middle East Stars and Stripes while he waited for the customers at the information desk in the terminal building. Cpl. O. H. Seals of Morristown, N. J., was looking over his shoulder.

"It happens to everybody," Seals said. "The big ones, too."

"But he was an awful smart guy," Patcheck said.

"Sure," Seals said. "But he's not the only smart guy. We've got others. Lots of them."

Sgt. Bob Bowlsma was reading the final item of the 5 o'clock newscast in the Armed Forces Radio Service station in Panama when Cpl. Reuben Diaz, the station's Spanish announcer, handed him the flash. GIs hearing it at supper in the mess halls didn't believe it at first. Then the station's phone started to ring. Sgt. Jim Weathers would pick it up and say, "Yes, it's true." "Yes, it's true," he said to each call. "Yes, it's true."

In Sydney, Australia, Sgt. Lloyd P. Stallings of San Antonio, Tex., said, "I came down here to have a good time, but now I don't feel so cheerful."

Pvt. J. D. Cotter of the Australian Army said, "Wish I knew more about this new bloke."

Outside the Grand Hotel in Paris, Pfc. Lester Rebuck, a medic from the 104th Division, said: "It was just like somebody socked me in the stomach when I wasn't looking. I just couldn't get it through my head that he was really dead. For my money, that guy was one of the greatest guys that ever lived. You can put him up next to Lincoln or Washington or anybody."

Source

“Bewildered GIs Stunned to Hear of the President's Death (GIs Overseas).” Yank, The Army Weekly, May 25, 1945.