जाएगा and जायेगा are interchangeable. Basically, in words where a vowel appears by itself in the middle of a word (like ए in जाएगा), you will have two vowel sounds one after the other. Some people don't like this and tend to insert a 'helper consonant' like य् (y) in the middle of two vowels to help in the pronunciation. So, instead of just 'e' in 'jāegā', they make it into a 'ye' in 'jāyegā'.
जाऊँगा is the first-person form and goes with मैं.