Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\}.$$ Let $\mathbb{A}$ denote $\mathbb{N}$ or $\mathbb{Z}$. Consider the following set (where $u \cdot v$ denotes the usual dot product): $$E(\mathbb{A}) = \{\|u \|^2 + |u \cdot v| \text{ such that } u,v \in \mathbb{A}^3 \text{ and } \|u \| = \|v \|\}.$$

Let $u= (a,b,c)$, $v = (b,-a,c) \in \mathbb{Z}^3$. Then $\|u \| = \|v \|$ and $\|u \|^2 + |u \cdot v| = a^2+b^2+2c^2.$ It follows that $F= E \subseteq E(\mathbb{Z})$.

In fact, Dickson's theorem extends to $E(\mathbb{Z})$, since Philipp Lamp shown below that $E(\mathbb{Z}) = F$ also (as an answer to what was Question 1 in a previous version).

The computation below suggests the following question (checked for integers less than $5936$).

Question 2: Is it true that $E(\mathbb{N}) = F \setminus \{ 5, 23, 29, 65, 167 \} $?

Application: this answer proves that the form $\| A\|^2$ covers every natural number for $A \in M_3(\mathbb{Z})$.
A positive answer to Question 2 would prove this result for $A \in M_3(\mathbb{N})$.

Reformulation of Question 2

Take $u=v \in \mathbb{N}^3$, then $\|u \|^2 + |u \cdot v| = 2 \|u \|^2$, so by Legendre's three-square theorem, $$2\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\} \subset E(\mathbb{N}).$$ So we are reduced to prove that $$2\mathbb{N}+1 \setminus \{ 5, 23, 29, 65, 167 \} \subset E(\mathbb{N}).$$

Now, as pointed out by Philipp Lampe, if $\|u \| = \|v \|$ then $\|u \|^2 + |u \cdot v| = \|u+v \|^2/2$.
Then Question 2 can be reformulated as follows:

Reformulated question: Is it true that, for $u,v \in \mathbb{N}^3$ with $\|u \| = \|v \|$, the form $\|u+v \|^2/2$ covers every odd number, except those in $\{ 5, 23, 29, 65, 167 \}$?

Computation

sage: L=[]
....: for a1 in range(50):
....:     for a2 in range(a1+1):
....:         for a3 in range(a2+1):
....:             x=a1**2+a2**2+a3**2
....:             b=0
....:             while b<50 and b**2<x:
....:                 b+=1
....:             for b1 in range(b+1):
....:                 bb=0
....:                 while bb<50 and bb**2<x-b1**2:
....:                     bb+=1
....:                 for b2 in range(bb+1):
....:                     bbb=0
....:                     while bbb<50 and bbb**2<x-b1**2-b2**2:
....:                         bbb+=1
....:                     for b3 in range(bbb+1):
....:                         if a1**2+a2**2+a3**2==b1**2+b2**2+b3**2:
....:                             n=((a1+b1)**2+(a2+b2)**2+(a3+b3)**2)/2
....:                             L.append(n)
....: l=list(set(L)); l.sort()
....: s=set(range(5936))-set(l)
....: S=[]
....: for i in s:
....:     f=list(factor(i))
....:     a=f[0][0]
....:     b=f[0][1]
....:     if a<>2:
....:         S.append(i)
....:     elif Integer(b).mod(2)==0:
....:         S.append(i)
....:     elif Integer(i/(2**b)).mod(8)<>7:
....:         S.append(i)
....: S.sort()
....: S
[5, 23, 29, 65, 167]