According to spell check it's chill, but not according to my conscience. Are 'Anyways' and 'Either ways' English terms? If so, can one replace 'Anyway' and 'Either way' with them? Thanks in advance.

They are nonstandard. Both are widely enough used to be endorsed by spell checkers, but I myself would avoid them outside of dialogue.

If I have doubts about a word, I don't trust spell checkers alone. I go to a site like dictionary.com:

http://dictionary.reference.com/browse/anyways
(Dictionary.com fails to find "either ways", but it does find "either way" as a set of web links, including one for the idiom "either way")

adding an 's' to either is generally considered poor speaking/writing... to me, the same goes for 'toward' but the 's' version seems to have been accepted by some...

if you wouldn't say/write 'anyhows' why would you use 'anyways'?

and 'either ways' makes no sense whatsoever!... 'either' refers to one of two whatevers... so, if it can't be 'either dogs' or 'either situations' what makes anyone think it can be 'either ways'?...

Are 'Anyways' and 'Either ways' English terms? If so, can one replace 'Anyway' and 'Either way' with them?

They are Americanisms I would say. I would drop the 's' as suggested, unless you happen to be quoting an American who is using it in speech.

They are considered nonstandard/substandard in the Americas, as elsewhere. Nor is their use limited to the Americas.

You shouldn't be using "anyway" and "either way" in your writing, unless you're writing dialog. They are generally used as filler and have no (or little) place in written language. "anyway" is just a colloquial way of saying "regardless."

Thanks for the advice, everybody. Apparently, anyways or either ways are not appropriate for use in writing, so I think I got the point.