Did Buddhism become a widespread religion primarily due to the political support of the Indo-Greek rulers of India who favored it in opposition to Jainism?
You have the right idea, but it is a little off base. I can explain. You probably already know about Alexander the Great and his conquests. Throughout the Afro-Eurasian continent, Alexander set up garrisons, or small military towns, to rule over a certain area and claim it for Macedonia. The soldiers in these areas set up their own societies, with Greek entertainment, agriculture, and language. After he died, however, his empire split, and his top military elites, or the generals, set up their own empires. One of these empires was the Seleucid Empire, ruled by Seleucus.
However, in the turmoil after his death, Chandragupta Mori founded the Mauryan Empire in what is present day India and pushed north into the Seleucid empire. After a few brief battles, the Seleucid and Mauryan empires agreed upon a treaty, and they participated in cultural and intellectual exchange. Keep in mind that the area that is now the Mauryan empire was once inhabited by the garrisons, so there was a large Greek influence in the Mauryan Empire, leading to a melting pot of cultures, beliefs, and ideas. New advancements were being made in roads and bridges, making transport through the Hindu Kush mountain range possible. These roads tended to lead directly into the Mauryan empire. At the same time, there were other nomadic influences like the Kushans and the Parthians, so the world was becoming more integrated. The Kushans promoted religious diversity, which will become more important later.
The overlap between the Seleucid and Mauryan empires affected new ideas and influenced one another. One main "bump" for Buddhism was Menander's belief that Buddha was a God. This evolved into Mahayana Buddhism, but keep in mind that only because of these Hellenistic/Greek influences was Mahayana Buddhism created. The idea of gods was contributed by the Greeks and adopted by Indians, creating an Indo-Greek culture. Mahayana Buddhism also believed in Bodhisattvas, or people who were ready to reach nirvana but held back to help others. When a Mahayana Buddhist died, they entered a rudimentary form of heaven, a "Buddha land", for those almost ready to reach nirvana. This convinced many less devoted individuals to become Buddhist because it didn't require the immediate renouncement of all desires, making it seem more enticing and possible. The Kushans promotion of Buddhism and Mahayana Buddhism expanded the Buddhist sangha, or followers. At the same time, there was a lot of literature being written about the Buddha, with ideas and intellectual exchange becoming more and more seamless.
So, to conclude, yes, Indo-Greeks were vital to the spread of Buddhism.