Using the articles database "New York Times Historical" (which many public and university libraries subscribe to), i did a search for the terms "vietnam" "soldiers" and "asylum". These are four articles that might be of interest to you, and different searches can yield different results as well.
DESERTER POLICY HELD TOO LENIENT
1) By ROBERT H. PHELPS Special to The New York Times
New York Times (1857-Current file); Mar 6, 1969; ProQuest Historical Newspapers The New York Times (1851 - 2005)
pg. 9
"...the case of a serviceman who had received three years' imprisonment and dishonorable discharge after seeking asylum in the Soviet embassy in Tokyo."
2) 9 G.I.'s Get Swedish Asylum
New York Times (1857-Current file); Dec 14, 1968; ProQuest Historical Newspapers The New York Times (1851 - 2005)
pg. 46
3) ASYLUM IN SWEDEN GIVEN 13 MORE G.I.'S
New York Times (1857-Current file); Jul 18, 1968; ProQuest Historical Newspapers The New York Times (1851 - 2005)
pg. 3
4) A.G.I. Assigned to Vietnam Defects to Cubans in Tokyo
New York Times (1857-Current file); Apr 14, 1967; ProQuest Historical Newspapers The New York Times (1851 - 2005)
pg. 3
"The Cuban Embassy here [Tokyo] said today that it had granted asylum to a Korean-born United States soldier assigned to a Vietnam duty and had rejected a demand from the Japanese Foreign Ministry that he be surrendered to the Tokyo Police."
