I see some problems with those two sentences. One problem is that they are comma splices, since "I don't get it" is a sentence in itself.
Another problem is with "eligible for". Specifically, "eligible for" is used with a noun (or a pronoun like "what"), but you can also say "eligible to verb" (i.e. "with infinitive" as in this definition). Thus, it would be grammatical to say:
Why would I be eligible for any gifts from them?
Why would I be eligible to receive any gifts from them?
Note: The "rules" for eligible may not be particularly obvious in the dictionary, but it becomes pretty apparent if you search a corpus such as the Corpus of Contemporary American English. Doing a search for
eligible for _v*, where
_v* matches verbs returns only 36 results, all of which are either "eligible for NounEndingInING" (e.g. "eligible for funding") or cases where there's a stranded preposition that refers to an earlier (pro)noun (e.g. "what we were eligible for is a loan").
While "how come I become eligible to receive..." is technically grammatical, the tense isn't appropriate. You could say "how come I became eligible to receive...", since presumably you became eligible in the past, before they shipped the gift. Or, you could say "how come I'm eligible for a gift?"
One last thing I'll note is that if you're looking for something informal you should replace "receive" with "get" (since the latter is more informal). Personally, I'd say:
I don't get it. Why'd I get a gift?
Note that "why'd" is nonstandard and ambiguous (as it could be either "would" or "did", not that it really makes a difference either way) but that's pretty typical of spoken English.