To favour non-buddhist teachings would be one of the 46 bodhisattva's secondary downfalls in Mahayana.
The most important Zen Buddhist master after Dogen, Hakuin, narrates in "Wild Ivy: The Spiritual Autobiography of Zen Master Hakuin" the story of contracting a severe illness brought about by the austerities of his most dedicated Buddhist practice, and being unable to get help from any sources, spiritual or medical. He was going to die, but then he was told there's a taoist hermit living in the mountains who may be the only one who can help. Hakuin made the desperate and difficult journey to the hermit's abode. The next chapter is dedicated to the taoist explaining to Hakuin the nature of his illness and the practices to cure it. Hakuin promptly got better and then completely well, and proceeded to teach, promote, and revitalize Zen for many decades, but he mentions that he practiced what the taoist gave him for the rest of his life. There was no "downfall" -- only prudence that benefited everyone involved.