'Choose' is a bit of a stretch, but of all the places in the Greater USSR at the time of German invasion in 1941, the Ukrainians were most (at first) receptive to German advances.

Bare in mind that the Ukrainians saw themselves as being subjugated by the Soviets, and had suffered enormously during Stalin's five-year plans. By contrast the Germans invading the USSR appeared to be the bee's knees. As an aside, it is considered yet another failing of the German regime that they did not capitalize on this goodwill. Had Hitler been more receptive to the Ukrainian population, they may have indeed found many ideological volunteers for their military. Instead, the Ukraine suffered the same fate as all other 'slavic' regions, and were subjected to horrors from German Einsatzgruppen. Needless to say, roving death bands wiping villages off the map, coupled with extensive looting for survival during the harsh winter, did nothing to endear the German occupiers to the Ukrainian population.

Now, to your question:

mother explained that he and his friends had to choose who they would fight for, and his friends told him that the German army takes better care of their members than Russia.

There is truth to this; in fact during the initial advances it was not uncommon for Soviets of various ethnicity to cross the lines and volunteer for the Wehrmacht. These men would come to be known as 'HiWis', an abbreviation for "Volunteer Helpers" and their role in combat would grow as German forces wore themselves thin and found themselves in more and more desperate circumstances. Needless to say, ethnic Ukrainians and Russians fighting in the German service - even if they did not cross over from the RKKA, could expect no mercy in the event of capture. Many more POWs would be forced into being 'HiWis' (making the term a bit of a farce), the less reliable they were, the greater the distance from the frontline they could expect to serve.

So, who would have made them choose? Well as said, it depends on the situation, there were as many HiWis pressganged into service by threadbare German units as there were HiWis who truly disdained the Soviet regime, or more often, felt that it had no hope of withstanding the German assault. The real issue of 'choosing' to fight with the Wehrmacht was that it was a dangerous choice; as reprisals against family back home became more and more of a possibility as the Soviets began to push the invaders out of their territories.

I am open to follow up questions, I'm just going to dash upstairs and grab some source books in the event they come :P. Hope I helped shed some light.

Edited: Minor elaborations

First off, are you sure you are talking about WWII? If you are referring to WWI, you might be referring to the fact that some Ukrainians were part of Austria-Hungary at the time.

That being said, in WWII, there were some Ukrainian troops amongst the Ostlegionen, or Eastern Legions. These were non-German troops recruited from populations in the east, and included in their number Polish (a number of whom were fascists), White and disaffected Russians, Ukrainians and Belorussians, Tartars and Cossacks, and even some Chinese and Koreans. These troops were almost entirely volunteers, although one would imagine that service in the German military would have been preferable to being a victim of the Einsatzgruppen.

There were also a group of Ukrainians who followed Stephan Bandera, who led the Ukrainian Independence Army, which first collaborated with the Germans against the Soviets, and then fought the Germans in 1943-1944, and then the Soviets again.