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.