This is an interesting question - an all the more interesting in this situation. The 106th inf division was severely mauled in the Ardenne offensive - and159th Inf Regiment actually replaced one of the original regiments in the formation. The 159th was a California National Guard Regiment assigned to the 40th Infantry Division. It remained in the US until March of 1945 when it was shipped to France.
Since it was replacing one of the 106ths original regiments it probably would have been authorized to wear the unit patch of the division. The question is - would they even want to? The 106th was a hard luck outfit and members of the 159th would have known that. The 106th with its new regiments later moved into Germany but was assigned to POWprocessing.
Units attached on a temporary basis would have probably kept their own insignia - if they didnt have a sleeve flash , if they wore anything they wore a corps level or army level insignia if they wore anything. The 159th had its own regimental insignia authorized - this would have been worn on the dress uniform in addition to a divsion/corp/army sleeve flash.