The line you quoted is the one seen in the North American instruction manual for SMB2, except it says "Cathy" instead of "Birdetta." People just claim it's what the JPN manual says because it's close enough. I'm not surprised that Google gave you that. A lot of those are similar to the words used, but Japanese is such a complicated language that a computerized translator will be rather poor.
Here, let's break it down.
Scan of the manualFirst, here is it Romanized:
"Jibun wo mesu to omoikonde kuchi kara tamago nado wo haku yatsu. Kya shii to yobu to kigen ga ii."
Jibun ~ self
mesu ~ female (animal)
omoikonde ~ compound verb, meaning: to be under impression that; to be convinced that; to imagine that; to set one's heart on; to be bent on; etc.
wo & to ~ necessary grammatical particles, giving the phrase the meaning "under the impression that self is female (animal)"
kuchi ~ mouth
kara ~ from
tamago ~ egg
nado ~ and such; etc.
wo ~ again, necessary grammatical particle
haku ~ vomit
yatsu ~ "dude"... informal term used when referring to a person (essentially, informal term of "person"). In the translator's experience/impression/observation, this is used only for males, and "dude" is actually a pretty close translation, if a little bit "euphemistic"... (I don't think "dude" is ever used in a derogatory fashion, but "yatsu" definitely can be). Highly doubt it's used in a derogatory way here.
Kya shii ~ "Cathy" or "Kathy" or "Cathie" or however you choose to spell it.
yobu ~ (verb) to call (or to summon, to call out, or to invite)
kigen ~ humour, temper, mood
ii ~ good
to & to ~ necessary grammatical particles... in this case, essentially meaning "if... then"
ga ~ another particle... essentially an equivalent of "is"
So there you go. If you put those words together in that order, it makes no sense, so you can construct it however you want as long as all the key words are there.