It would be interesting to know where you got this idea of "eijaa" as an exclamation. As far as I know, it's "heijaa". (Of course there can be regional variation in this but it just doesn't sound like a natural "happy exclamation" since "ei" means "no". On the other hand, "jaa" can mean "yes", so perhaps it's +/-.) Not that I have any other theory for the name's origin, but all the same.
'Eijaa' is only used in old hymns, it comes directly from the Latin word 'eia', but it's not used in real Finnish. 'Heijaa' is also rare on its own and it comes from 'heia' which is an alternative on 'eia'.