Both expressions are correct Standard Portuguese, both in Brazil and in Portugal. But "Eu chamo-me" is preferred in Portugal, and very unusual in Brazil. In colloquial Portuguese, only "Eu me chamo" is used in Brazil.
In Standard Portuguese, the expression "Me chamo", without the pronoun, is deemed incorrect (if there are no other words in the sentence before "me"). But it is very usual in colloquial Brazilian Portuguese¹. I suppose that "Chamo-me", with no pronoun (which is correct Standard Portuguese) is also very usual in colloquial European Portuguese.
If there are other words, especially adverbs, before "me", then a proclitic construction might be preferable, and even mandatory, in Standard Portuguese (and on both sides of the ocean):
Eu não me chamo João.
*Eu não chamo-me João.
Ninguém sabe que me chamo João.
*Ninguém sabe que chamo-me João.
Felizmente me chamo João.
Felizmente chamo-me João.
- This is true of all verbs, but especially of those forms where the last pre-vocalic consonant is the same as the initial consonant of the clitic, as is the case in "chamo-me" (or "chamaste-te") – the repetition of the sound is awkward for Brazilian sensibilities. "Chamava-me" or "chamas-te" would be marginally better received.