Learn Russian: Я болен: то мёрзну, то потею.

It translates to 'now ..., now ...' or 'first ..., then ...'. It is used for alternating actions that keep switching back and forth. Because the speaker keeps switching between freezing and sweating, we use the imperfective verbs мёрзну and потею to show these are repeated, ongoing actions.

Why use the short form болен instead of the normal adjective больной?

Болен is a short-form adjective used to describe a temporary state: 'I am sick right now.' If you used the long form больной, it sounds more like a permanent trait, meaning 'I am a sick person' (and in casual speech, it can even mean 'crazy').

Why isn't there a я (I) before мёрзну and потею?

Since you already started the sentence with я (I), you don't need to repeat it. Russian frequently drops pronouns when the context is obvious, and the verb endings (-у in мёрзну and -ю in потею) already make it perfectly clear who is doing the action.

Could I use this то ..., то ... structure to talk about being sick yesterday?

Yes! You would just change the verbs to the past tense: вчера я то мёрз, то потел (yesterday I was freezing, then sweating). The structure remains exactly the same, and the verbs stay imperfective to show the repeated switching in the past.