The Huns were traced to a nomadic tribe in Central Europe (the Steppes near the Black Sea). It is still uncertain whether the Huns in Europe were the same as Xiongnu (which sometimes are also called Huns) [see Encyc. Britannica].
Huns in Europe appeared around 370 AD, while the presence of Xiongnu was felt in China during Qin dynasty (221 BC). The result of conflict between Han dyasty and Xiongnu was a division of Xiongnu. Part of the Xiongnu tribe was "sinicized" (Hanized?), as recorded in history about Wang2 Zhao1 Jun1 and Han Wudi. The others were driven away. It is not surprising that after several hundred years, they actually showed up in Europe.
Hungary, which obviously is derived from the word "Hun", has a language of Uralic origin. It is quite certain that Hungary had heavy influence by the Huns. Interestingly, a Hungarian friend told me that Hungarians put their family first when they address people, which is distinctly different from other European culture, but similar to Chinese culture. Whether this is the influence of the original Huns or the Mongolian occupation later in 1200 AD is uncertain.
Huns are significantly different from Hakka in their cultural behavior. Although both Huns and Hakkas are migratory, Huns never settled in one place. They kept moving, conquering and moving. Huns mainly made their living by snatching from the conquered while Hakkas are agriculturally based and self-sufficient. Huns were illiterate and had no idea about civilization and knowledge preservation, while Hakkas have a tradition of strong emphasis on education and intellectualism. These two cultures are totally dissimilar and incompatible. Huns finally disappaered and was integrated with Europeans without a trace of their original "culture".
Xiongnu in China also intermarried Han people. During the downfall of West Jin dynasty, the Han people cross the yangtze River and settled in southern China, bringing with them some Xiongnu soldiers and servants. While Xiongnu descendents established "Han" Kingdom in the north, gradually became sinicized. Han Kingdom was destroyed by Zhao kingdom (Jie2 ethnic group), which was in turn destroyed by Han people again.
If Hakka were actually sinicized "non-Han", then Hakka migration from north to south would not be "fleeing" the "northern foreign invasion" to "preserve" their own culture. Intead, Hakkas would have to be the actual "invaders" from the north trying to spread their own culture to the south. However, how a non-Han minority could preserve the Han culture better than the true Han people would be very difficult to explain. And it would be even more difficult to explain why the poems in Shijing (The Book of Poems) popular in the Chunqiu-Zhanguo period (pre-Qin) rhymes better with Hakka than Mandarin. Xiongnu although had attempted invasion of the northern kingdoms during the Chunqiu-Zhanguo period, they could hardly have had major settlement in "China" prior to Han dynasty.
The "theory" on Hun origin of Hakka was based on very fragmentary blood typing and DNA analysis done by Japanese and Russian researchers.
(I believe that the Hakka are of mixed race, Asian and Cacuasian.)