The Chinese character for "worms," "insects" or "bugs" is 虫 [chóng]. This is a general name for creatures that are very small in size, so it is not a scientific term meant to differentiate among worms, insects, or bugs.
In very ancient times, the pictograph was drawn like this:
Over time, it was standardized to be 虫。
Worms or insects are not particularly intelligent (with the obvious exception of ants), and the Chinese sometimes denigrate people for being slow-witted or muddle-headed by calling them 糊涂虫 [hú tú chóng] - "confused worms."
Small creatures such as bugs have short life-spans. There is the Chinese saying 夏虫不可语冰 [Xià chóng bù kě yǔ bīng] - "You cannot discuss ice with summer bugs."
Worms that eat their way through materials such as wood or books are called 蛀虫 [zhù chóng] - “tunneling worms." This term is often used to refer to people who stealthily damage an institution to benefit themselves, such as corrupt officials.
For the most part, people do not care much about worms or bugs, and it's hard to think of any Chinese term that compliments them. Insects are somewhat different. Some Chinese keep crickets as pets. In fact, 花鸟鱼虫 are commonly known as four homely hobbies in China. Grasshoppers also frequently appear in traditional Chinese paintings as a fixture of the natural world. Below is a piece done by a famed artist, Qi Baishi, who lived in the late 19th and early 20th century.