There are mixed opinions over how closely the German armed forces were tied to the Nazi party, but there is a quantifiable difference in the formal connection between party and army between the two groups you mention.

The Wehrmacht served directly under the Nazi party itself rather than the German government as a whole. In this regard it is similar to the People's Liberation Army in China today. The Wehrmacht Heer was established in force by a conscription law enacted in 1935 by the Nazi's. Before this there had been a long tradition of armed forces in Germany but that was interrupted by the defeat of the country in the first world war and their subsequent disarmament. This allowed the Nazi's when they reinstated conscription, and generally took power in the proceeding years, to reshape the institution as a party apparatus rather than an apolitical entity. The top generals at this time all swore personal oaths of loyalty to Hitler. This doesn't mean they were actively loyal to him, and indeed there were major plots against Hitler originating from the army, but the Wehrmacht was defined as a group which pledged loyalty to the Nazi's.

The US armed forces serve under the federal and state (national guard) governments and do not have a direct political affiliation. Their oath is to uphold the constitution above all else, including the commander in chief. It is also important to consider that the American armed forces serve a 2 party system instead of one. In a one party system support of the government (assuming armed service implies support) is more easily equated to support of the party, but with two opposing parties sharing power it is more difficult to say of a person "he supports the government so he must be a democrat."

Also keep in mind that though the Wehrmacht started as a volunteer force eventually they were forced to recruit nearly any man physically able to fight due to massive manpower shortages. The current united states armed forces are all volunteers of varying degrees. Figures from the early Wehrmacht might more accurately compare with the US Armed Forces today because of this distinction, but considering you ask about defection and the Red army that implies a later point in the timeline, when German soldiers were often forced to serve.

I don't have a source on hand for the political affiliation of German soldiers at that time but if I can find one I'll update this post. I'm also not certain on the statistics for defections vs. Surrenders during the war and I'm not sure how accurate any source on the matter would be considering the political benefits of boosting or downplaying the numbers there.

Is it known how many German soldiers opposed the German government and tried to leave or even join the Red Army?

Defection was quite common on both sides. Germans weren't accepted in military units on Soviet side (unlike Czechs, Slovaks or Romanians for example) but sometimes were used in a similar way Russian POW's were used by Germans - Hiwi (usually as drivers for captured German trucks). There's no statistic on that matter, because using Germans in such way was strictly forbidden, but you can get general idea from some memoirs of Russian soldiers.

All in all, there was no way for German soldier to join the Red Army, but defectors usually were treated better than POW's and usually they've got some jobs in propaganda units.