The answer to this question can best be described as complicated and multi-faceted.
First of all, Bosnia and its Muslim population were not under direct German rule. When the Germans invaded Yugoslavia in April 41, their priority was to establish an occupation that would use as little man power and resources as possible because of their preparations to invade the USSR. Due to this factor and also due to Italian pressure for Mussolini hoped to excert a lot of influence in the Balkans, Bosnia was given to Croatia under the rule of Ante Pavelić's Ustaša (or Ustasha if you prefer English transliteration).
The Ustasha has been described by historian Alexander Korb as the radical product of the disintegration of the Habsburg empire. Racialist nationalists, they were a conspiratorial terrorist organization that had fought the Yugoslav state and the Serbs in particular. When they took over in Croatia and subsequently founded the Independent State of Croatia (Nezavisna Država Hrvatska, NDH) they were in essence the second choice of the invading Nazi regime because the conservative Agrarian party had refused to collaborate. The Ustasha, at the time of their takeover a group of about 3000 people, quickly set to put their policies against Jews, Roma, and Serbs into bloody practice. They for example dislocated about a third of the Serb population of Croatia, while another third was to be killed, and the final third to be converted to Catholicism.
Now, where do the Bosnian Muslims figure into this. This is where it gets a bit complicated. The Ustasha wanted to use the Muslims as pawns in their policy of forced expulsion, murder and forced conversion against the Serbs. They declared that the Bosnian Muslims, which they initially regarded as »Muslim Serbs« to be Croats and started spreading hatred between the Serbs and the Muslims in Croatia, in part by drafting Muslims into their security forces, in part by adopting Muslim symbolism such as the Fez. This tactic worked to a certain extent, and the Serbian nationalist resistance in Croatia and Serbia, the Chetniks, started not only attacking the Germans and Ustasha but also Muslim communities.
At the same time, a considerable number of Bosnian Muslims were also targeted for imprisonment and to be killed by the Ustasha because they were so-called »gypsies«. Roma in Croatia had often adopted the Muslim faith and either settled down or continued their Roma lifestyle. Since the Ustasha targeted so-called gypsies as part of their policies of racialist cleansing, the did indeed imprison and often kill in a very brutal and gruel way, considerable amounts of Muslims.
Basically, a lot of Bosnian Muslims were caught between being victimized by the Chetniks because they believed them to be Ustasha and the Ustasha for they believed them to be so-called gypsies. This motivated quite a few of them to join the Partisans, who had a policy of being open to all nationalities (in 1941 e.g. people from Bosnia and Herzegovina were the third largest group in the Partisan movement after Serbia proper and Montenegro, by 1945 about a 100.000 of them fought with the Partisans). But this open was also not necessarily open to all since the Partisans, being communists, were not exactly regarded highly withing faithful communities.
This problem of being caught between the fronts was compounded further by the lack of central leadership within the Bosnian Muslim community. Bosnian Muslims had not been a nationality within Yugoslavia before the war, like it became later under Tito. The only thing resembling something of a political institution (aside from those who fought with the Partisans) was the clergy. And in this group reactions varied.
On October 12 1941 108 notable Muslim citizens of Sarajevo declared their opposition towards the Ustasha policy towards Serbs. This came in part from them feeling like they were caught between the fronts, partly because of their horror towards the brutal treatments of Serbs by the Ustasha. In this resolution, they condemned the cruelty of the regime and highlighted the persecution of many of their own. Soon, similar resolutions followed from other Muslim communities. Of course, this didn't really bring about the desired effects and the signatories were subsequently threatened with imprisonment in a Concentration Camp by the Ustasha. This lead to a variety of responses.
Those who were liberal or inclined towards the left began seeking their salvation with the Partisans, those who were not, either started organizing armed units of Muslims as a self-protection force for Muslim communities (Milovan Đilas, one of Tito's confidants mentions them in his autobiography of the war as fierce fighters who often attacked pretty much everyone who came near Muslim communities). And a third group comprised of conservative clerics formed the Committee for National Salvation in August 1942. This committee petitioned the Nazis to take over Bosnia and basically replace the Ustasha as rulers of Bosnia. An interesting detail about this is that most of them being rather old at the time, ask for the return of the Austrian administration of pre-WWI times. Apparently, they tended to remember the Austrian-Hungarian rule over Bosnia rather fondly compared to the Kingdom of Yugoslavia and the Ustasha.
This opportunity was also seized upon by the Nazis who exploited the Bosnian Muslims' need for security from the Ustasha and the Chetniks to recruit a Bosnian Muslim Waffen-SS Division »Handjar« among them. Handjar was far from the success the SS-Hauptamt (SS Main Office) had hoped it would be, especially in terms of recruits. Also, its members were promised to be only used in Bosnia, so when the Division was sent to France in order to be trained for use in the USSR, a mutiny broke out and a rather considerable number of people deserted from the unit.
Summing up, treatment and response of the Bosnian Muslims in WWII varied and is hard to pin down. While some happily collaborated with the Germans and Ustasha, a considerable number also fell victim to their or the Chetnik's policy. While some joined the Partisans, others organized themselves and again others sought their luck with the Germans. However, this is not exactly untypical (except the seeking refuge with the Germans part) for an occupied population in Europe.
Sources:
Enver Redžić: "Bosnian Muslim Policies". In: Bosnia and Herzegovina in the Second World War, London 2005.
Robert J. Donia: "The multiple roles of Sarajevo's muslims". In: Sarajevo : a biography, Ann Arbor 2006.
Alexander Korb: ‘Ustaša Mass Violence Against Gypsies in Croatia, 1941/42,’ In: Anton Weiss-Wendt (ed.) The Nazi Genocide of the Roma: Reevaluation and Commemoration, New York, in print.
Alexander Korb: ‘Understanding Ustaša Violence’. In: Journal of Genocide Research, 12 (2010), 1–18.
Alexander Korb: ‘Nation-building and mass violence: The Independent State of Croatia, 1941–45’, in Jonathan C. Friedman (ed.), The Routledge History of the Holocaust (Routledge, New York, 2011), 291-302.
Martin Broszat: Der Kroatische Ustascha-Staat, 1941–1945, Stuttgart, 1964.
the works of Jozo Tomasevich.
Looks like I have been subject to Balkanism.
The sources you cited before, they aren't in English, are they?