What kind of meaning does "なきゃともがいてる" convey in both of the sentences in one of the panel above?
- なきゃ＝なければ(ならない) : must
- と makes a clause like と言う
- もがいている : is struggling
A simple way of understanding this would be to add '思って' after 'と'. Thus
生きなきゃともがいている = 生きなければならないと思ってもがいている
struggling because he thinks he must live
struggling because he thinks he must go forward.
Another way is to take this construction as a mix of
- ・・・しようともがく: struggle / try hard to [do], and
- なきゃ : 'must', adding the sense of duty or obligation
so the above sentences mean 'telling himself to live on' / 'trying to persuade himself to go forward'.
The both ways shouldn't make a lot of difference, but I suppose the latter is a better way to understand them (based on my understanding of the English version.)
A side note:
なきゃ does not always mean 'must'. See e.g. this post. Unlike your example, such なきゃ (as in the title of the post) can be replaced only by なければ, and not by なければならない. (Both can replace なきゃ in your example)