All the natural numbers are of form $x^2+y^2+z^2+w^2+xy+zw$ Is there any natural number that cannot be written as $x^2+y^2+z^2+w^2+xy+zw$, where $x,y,z,w$ are integers? If there is none, how do I prove that all the natural numbers are of form $x^2+y^2+z^2+w^2+xy+zw$?
$x^2 + xy+ y^2$ represents exactly the same numbers as $u^2 + 3 v^2.$ You are asking whether $$ s^2 + 3 t^2 + u^2 + 3 v^2   $$ is positive universal. It is. 

More: a variation of quaternion multiplication shows that, when this form represents two numbers, call them $m,n,$ then it also represents $mn.$ Meanwhile,
$$ s^2 + 3 t^2 + u^2  $$ is a regular ternary form, in particular it represents $1,2,3,$ and all numbers not divisible by $3,$ which includes all other primes. From the multiplication property, it follows that all (positive) numbers are represented. 

For information about what numbers are represented by $s^2 + 3 t^2 + u^2,$ see the entry for $1,1,3$ in Dickson_Diagonal_1939.pdf at TERNARY. Acomplete proof is given as Theorem 89, pages 97-99, of Dickson's book.
By the $15$ theorem, it suffices to check all naturals up to $15$. The margin is too small to fit my proof so the task is left to the reader.