Following the link from Steve Bird, one finds the biographies of 28 expedition members.
In Timothy McCarthy's biography:
On returning to England after the expedition, McCarthy joined the Royal Navy Reserve as a Leading Seaman as his service in the First World War. On Friday 16th March 1917, only three weeks after returning from South Georgia, and at the age of 28, he was killed in action at his gun post on board the S.S. Narragansett, on his first day under enemy fire. The ship had been torpedoed between the South West of Ireland and the Scilly Isles. He went down with his ship along with all other 45 hands, he was the first of the Endurance expedition members to die.
In Alfred Cheetham's biography:
On return from the Antarctic he enlisted in the Mercantile Marine and while serving as second Officer on the S.S. Prunelle on Thursday 22nd August 1918, at the age of 51, his ship was torpedoed in the North Sea by a German U-Boat. Alf went down with the ship, he was the second of the men of the Endurance to be killed in the Great War.
That page currently lists biographies for 28 of the crew (including Shackleton).
Now, going to The History of Parliament one finds
Of the 53 surviving members of the expedition, three were killed in the war and another five were wounded.
An issue here is that Wikipedia and other sites from Google list 28 total crew (including Shackleton), matching the number of biographies given on the CoolAntarctica site. This brings The History of Parliament site into question.
The only two listed as killed in World War 1 are those listed above. Huberht Hudson (per the bio, Huberht is an older spelling of Hubert but is what Hudson used), the navigator, was killed in action in World War 2 which may have led to the incorrect listing of 3 killed in action. That still does not explain the 53 crew members vs 28 listed elsewhere.
Going through all the biographies on Cool Antarctica, one finds that of the 28 there is no record of World War 1 service for only 5 of them. The majority saw service in the Navy or the Merchant Marine.
Of possible interest:
Frank Worsley (Captain of Endurance) commanded Q ships, sinking 3 U-boats and receiving the DSO.
Dr. James McIlroy, surgeon, badly wounded at Ypres.
James Wordie, scientist, badly wounded at Armentieres.
Charles Green, cook, wounded on HMS Wakeful.
Walter How, able seaman, blinded in one eye by German mine while serving on a Merchant Navy vessel.
John Vincent, able seaman, survived being torpedoed.
William Bakewell, able seaman, an American who served in the British Merchant Navy and survived two sinkings.
So, based on my review, only two of the crew of the Endurance died in World War One, and one in World War Two.