Everyone have their own dream/Everyone has his or her own dream?

Everyone as a subject is singular, so I think the correct sentence would be the one with has. The mistake people make with have is probably because everyone stands for all the people, and with people you would use have.

this sentence "Everyone has his or her own dream." should be the correct one.

It's ok, but it would've been better if you had used the instead of this because it's a rule when you mention something already, that you use "the" the second time you mention it.

and also maybe: I have searched for these two sentences instead of: "I search of these two sentences..."

Otherwise, your English is pretty good. :)

Everyone Has A Dream

Both everyone has his own dream and everyone have their own dream are correct. Because, has is used for singular and have is used for plural meaning. Therefore, both of u are correct.

"Everyone has his or her own dream" is perfectly correct. However, increasingly people now tend to use "their" instead of of "his or her" to make it more succinct. While incorrect, it is very common and is understood to mean "his or her".

An alternative would be to be "Everyone has his dream" -- but this risks sounding sexist!

A better way might be to put "Everyone has a dream" which bypasses the issue altogether.

PS: I don't think you need the "own" in. One's dream is one's own dream.