Reasons for Germans often being encountered among mercenaries in different armies are roughly the same as reasons why, say, Japanese were so popular as mercenaries in Asia: a lot of fractured lands in constant turmoil are a good breeding ground of excessive amount of men who are quite good at waging warfare and don't really want to earn their gold in any other way. Big amount of different states, cities and provinces which formed present day Germany meant that any side could summon any amount of mercenaries from anywhere and make them fight for their cause since these states would often quarrel with each other and could theoretically support many participators in many conflicts. It wasn't unusual for Germans on one side to face Germans on the other side while fighting in very remote areas. For example, in the era of Great Italian Companies where big swathes of mercenaries roamed Italian city-states, Germans actually supplied majority of soldiers in 14th century who could and most likely did take up arms against each other because both sides in conflict provided payment attractive enough to risk your life even against people you're familiar with.
By the 16th century Germans are already very common and popular choice of mercenaries as poor control over many counties and principalities of Germany provided much needed freedom in recruiting enough unscrupulous heads for every taste. Before the Italian War of 1542-1546 between France and Holy Roman Empire erupted, both sides went of remarkable recruiting campaign spendind a lot of money to lure as many swords for hire away from their competitors as possible. For example, king Francis I tried to persuade duke of Kleve to "lend" some cavalrymen since duke himself had a dispute with the emperor because of the Heldern principality.
It's also woth noting that, despite all their virtues, many authors criticized Germans as being unreliable:
... But when it comes to sobriety and military discipline, they are no match for Spanish or Italians who are much needed to stand among the German ranks...
This was probably a reason why some countries found Germans to be excellent trouble solvers and why some preferred to apply for other mercenary groups. I can reliably speak only on the matter of Eastern Europe, but at least in Russia French, Dutch and Scottish mercenaries were also quite popular in late 16th to late 17th centuries, sometimes probably even more so than Germans.