Learn Hardcore Japanese: watasi ha kenkou no tame ni tamago wo tabemasu.

は is the topic marker, indicating that 私 ("I") is what the sentence is about. It sets 私 as the theme. が is the subject marker often used to introduce new information or to emphasize the subject. Here, since you’re simply stating what you do for health, は is the appropriate choice.

Why is there a の between 健康 and ため?

In Japanese, の connects two nouns to show possession or qualification. 健康のため literally means "the sake/purpose of health." Without の, the phrase would be ungrammatical because ため needs a modifier before に.

What does ために mean in this sentence?

The phrase ために attaches to a noun or a verb dictionary form to express purpose or reason. Here, 健康のために means "for the sake of health" or "in order to be healthy."

Why is に used after ため?

に is part of the fixed expression ために, which marks purpose. It turns the noun ため ("sake/purpose") into a directional or goal-oriented phrase, equivalent to "for" in English.

What is the function of を after 卵?

を is the direct-object marker. It tells us that 卵 ("eggs") is what you are eating. In other words, it marks the thing that receives the action of the verb 食べます ("eat").

Why is the verb 食べます used instead of the plain form 食べる?

食べます is the polite, non-past form of 食べる. It’s commonly used in formal or courteous contexts. If you’re speaking casually to friends or family, you could say 食べる instead.

Can 私は be omitted in this sentence?

Yes. Japanese frequently drops the topic when it’s clear from context. You can simply say:
健康のために卵を食べます。

Can we change the word order to 卵を健康のために食べます?

Absolutely. Both 健康のために卵を食べます and 卵を健康のために食べます are grammatically correct. The meaning ("I eat eggs for my health") remains the same, though the emphasis slightly shifts depending on which element you place first.