Two of the three teachings in China are native - Daoism and Confucianism - however, my understanding is that Buddhism came into China in the middle of the Han and would become a large part of Chinese culture in the Tang Dynasty. What record do we have of Confucian and daoist reaction to Buddhism?
What record do we have of Confucian and daoist reaction to Buddhism?
There is a significant amount of documentation regarding the reaction of both Daoists and Confucianists.
Confucianists generally had the following complaints:
Daoists also complained about Buddhism being a foreign religion and had significant jealousy when Buddhism received more respect and patronage than they did. Occasionally they would have short term victories where their official status would be raised higher than that of Buddhism's at the imperial court. There would also be some periods were Daoists were allowed to conduct more ordinations than Buddhists.
A good source for more information: "Buddhism under the T'ang" by Stanley Weinstein
Similarities between Daoism and Buddhist teachings were recognized relatively early on. Popular religious beliefs from the Han-era- like the moral outcomes of good and evil actions, the favor of Heaven, etc.- were broadly compatible with karma and Buddhist theories about moral restraint and proper behavior.
A syncretic belief emerged that after departing from China, Lao Tzu had traveled to India and became known as the Buddha. This made it easier for Buddhism to gain acceptance in Chinese society- while the orthodox Confucians disdained it as a foreign teaching, seen as a kind of 'foreign Daoism' it could be incorporated into the existing religious order more easily. Pre-Confucian ritual practice and Taoist thought were combined in the Han era in a figure called Huang-Lao that combined the Yellow Emperor and Lao Tzu, and Buddhism was regarded by some Chinese thinkers as an alternative embodiment of the Huang-Lao teachings that seem to have been at the center of relatively early Han religious thought.
Taoism also had a well-established glossary of technical language for talking about metaphysical phenomena, many of which were employed to translate concepts from Buddhism. If you look at the language of Taoist texts like Zhuang Tzu, you will see many instances of language that sounds very similar to later Chinese expressions of Buddhist thought.
There was probably some Buddhist presence in China before the earliest recorded mentions, but Confucians controlled the official chronicles and preferred not to make note of foreign beliefs. Later, forgeries were introduced by both sides- by Buddhists to suggest an older Chinese pedigree for Buddhism, and by Confucianists to give the impression that such a significant topic hadn't been unknown to Confucius.
The Chinese critiqued Buddhist practices that were seen as particularly strange or objectionable (corpse contemplation, for example, wasn't readily accepted) but saw Buddhist dietary practices and breathing exercises as being very similar to comparable Taoist practices. Buddhist monks operating in China responded by preferring to translate texts that covered practices that the Chinese more readily accepted, which is thought to be one reason why texts explicitly dealing with meditation are more common in the early Chinese tradition than in the Indian tradition, where meditation was often more likely to be confined to commentaries and direct teacher-to-student instruction.
Influence from Buddhism probably encouraged Taoists to expand their pantheon, since the Buddhism that came into China from India probably already had a large collection of gods, Buddhas and bodhisattvas. It may have also encouraged the production of a more structured Taoist written canon, in order to have a larger body of writing comparable to the Buddhist canon.
Source: Kenneth Ch'en, Buddhism in China: A Historical Survey