Firstly, as a native speaker the line is clearly "Are you gonna go my way?".
Often in songs, one syllable will correspond to one note. In this case, both "are you" and "gonna" correspond to one note each while having two syllables. This makes the singing twice as fast as the other words ("go my way").
"are you" is pronounced /ˈɑ.jə/ because the word "you" is not emphasised. Normally in American speech "are" is pronounced /ɑɹ/, but because he's singing it so fast the /ɹ/ ("r") gets dropped, which is normal in parts of the English speaking world with non-rhotic accents. American accents are mostly rhotic, which means they always pronounce their "r"s.
/j/ ("y") is a semivowel and in some words is almost indistinguishable from /i/ ("ee"). "either" is pronounced /ˈaɪ.ðə(ɹ)/ ("eyether") or /ˈiː.ðə(ɹ)/ ("eether"). You might be hearing /ɑj/ as /aɪ/ or /iː/. With the /ə/ (no standard spelling, ~"a") at the end, you might hear "eye-a"/"a-ee-a" and think it's "either".
"gonna" is slang for "going to". "going to" is pronounced /ˈɡɔ.ɪŋ.tə/ because the word "to" is not emphasised. In "gonna", /ɪŋ/ ("ing") is reduced to /n/ ("n") and /tə/ ("to", ~"ta") is reduced to /ə/ (no standard spelling, ~"a") giving /ˈɡɔ.nə/.
I can't explain why you seemingly don't hear this word, since hearing "either" for "are you" seems completely explained as above.
Since your profile says you're in Galicia, Spain, I will assume you speak Galician. According to Wikipedia, Galician phonology has no consonant /j/, although /j/ is found in diphthongs (a two vowel sound starting on one vowel and ending on the other) such as /aj/ ("caixa"). Galician also has /θ/ ("macio", "cruz"). Since /j/ is always the last sound of a Galician diphthong, you are used to hearing a consonant after it.
In English, /j/ is usually a consonant so is almost always followed by a vowel. Since you want to hear a consonant, I think you hear a /θ/ where none exists, although the sound in "either" is actually /ð/. The difference between [θ] and [ð] is that [θ] is unvoiced and [ð] is voiced. Wikipedia doesn't list /ð/ as a phoneme in Galician, although it does give a few sample words with /ð/ in them ("corda", "couberdes", etc.). It's possible that in Galician [ð] is an allophone of [d].
What is sung is /ˈɑ.jəˈɡɔ.nə ɡoʊ maɪ weɪ/, which is quite different from /ˈaɪ.ðə ɡoʊ maɪ weɪ/ and /ˈiː.ðə ɡoʊ maɪ weɪ/.