I think I would call this very useful, well-made thingy a plastic cutlery basket, not a health chopsticks cage.
The word “health” is not typically used as an adjective, except to describe a few very abstract things like insurance. The adjective “healthy” is typically used to describe things you eat, drink, or do that are beneficial to your health, and would be just as inappropriate, though probably funnier. I do not think the product name needs a word corresponding to “sanitary” or “hygienic” because those words connote disposable things or substances whose purpose is to clean the body or disinfect something.
To my ears it always sounds strange to use plural nouns as adjectives.
A “cage” is totally enclosed, and usually has something alive inside, whereas this thing is open on the top.
便万家 (biànwànjiā) means “convenient 10,000 home”
品位生活 (pǐnwèi shēnghuó) means “quality life”
餐具收纳 (cānjù shōunà) means “tableware hold[er]”
卫生筷子笼 (wèishēng kuàizi lóng) means “hygienic chopstick container”
MADE IN CHINA
at Taizhoucity Huangyan Area Xidelai Plastics Factory