First question is a well known problem studied by Euler and some variants of it still out of our reach . For every positive integers $(x,y,z)$ let $P(x,y,z)$ denote the predicate: $(xy+1)(yz+1)(zx+1)$ is a square and not all of $(xy+1)$, $(yz+1)$,$(zx+1)$ are squares
Proof by infinite descent:
we well prove by the infinite descent that if there exists a solution $(x,y,z)$ such that $P(x,y,z)$ is true than there exists another solution $(p,q,r)$ such that $P(p,q,r)$ is true and $p+q+r$ less than $x+y+z$.
Let $(x,y,z)$ be tuple of positive integers such that $P(x,y,z)$ is true and $x\leq y \leq z$, consider the two positive integers $s_{+}$ and $s_{-}$ defined by :
$$s_{\mp}=x+y+z+2xyz\mp\sqrt{(xy+1)(yz+1)(zx+1)} \tag 1$$
This integers verify :
$$x^2+y^2+z^2+s_{\mp}^2-2(xy+yz+zx+xs_{\mp}+ys_{\mp}+zs_{\mp})-4xyzs_{\mp}-4=0 \tag2$$
And we can also prove the following important identities (they are basically the same):
$$
\begin{align}
(x+y-z-s_{\mp})^2&&=&& 4(xy+1)(zs_{\mp}+1) \tag 3 \\
(x+z-y-s_{\mp})^2&&=&& 4(xz+1)(ys_{\mp}+1) \tag 4 \\
(x+s_{\mp}-z-y)^2&&=&& 4(yz+1)(xs_{\mp}+1) \tag 5
\end{align}$$
we can use this identities to proof that $P(x,y,s_{\mp})$ holds, by multiplying $(4)$ and $(5)$ you will get that $(xy+1)(ys_{\mp}+1)(xs_{\mp}+1) $ is a square and not all of $(xy+1)$,$(xs_{\mp}+1)$ , $(ys_{\mp}+1)$ are square (as we have $(xz+1)$ is a square iff $xs_{\mp}+1)$ is a square and $(yz+1)$ is a square iff $(ys_{\mp}+1)$ is a square using $(4)$ and $(5)$).
Now the important par is to prove that either $x+y+s_{+}<x+y+z$ or $x+y+s_{-}<x+y+z $ this is equivalent tp proving that $s_{-}s_{+}<z^2$ which is true because :
$$s_{-}s_{+}=x^2+y^2+z^2-2(xy+yz)-4=z^2-x(2z-x)-y(2z-y)-4<z^2 $$
(remember that $x\leq y \leq z$)
Reference:
When Is (xy + 1)(yz + 1)(zx + 1) a Square?
Kiran S. Kedlaya
Mathematics Magazine
Vol. 71, No. 1 (Feb., 1998), pp. 61-63
Second Question:
The following Pell equation have infinitely many solutions :
$$x^2-3y^2=1 $$
For any solution $(n,m)$ of this Pell equation, let $(a,b,c)=(2n-m,2n,2n+m)$ then $ab+1,bc+1,ca+1$ are all squares.
It is easy to see that if $(z,u,w)$ is a primitive pythagorean triple, then $w$ is odd and either $z$ is odd and $u$ is even or $z$ is even and $u$ is odd. WLOG suppose that $u=2x$ is even, so $z$ is odd. Then
$$u^2=4x^2=w^2-z^2=(w+z)(w-z).$$
It is plain that $\gcd(w+z,w-z)=2$, so $y_1:=(w+z)/2$ and $y_2:=(w-z)/2$ are relatively prime positive integers. Thus
$$x^2=y_1y_2,$$
and therefore $y_1$ and $y_2$ are relatively prime divisors of $x^2$. This implies that $y_1$ and $y_2$ are perfect squares, so write $y_1=a^2$ and $y_2=b^2$. Hence,
$$u=2x=2ab,\quad z=y_1-y_2=a^2-b^2,\qquad\text{and}\qquad w=y_1+y_2=a^2+b^2.$$
This proves that every primitive triple has the form $(a^2-b^2,2ab,a^2+b^2)$, as wanted.
Of course, the condition $\gcd(a,b)=1$ is needed, otherwise $(a^2-b^2,2ab,a^2+b^2)$ won't be primitive.
Best Answer
Since $a^2, b^2, c^2$ are in arithmetic progression, $a^2+c^2=2b^2$. If $a$ is even, then so is $c$, so $4 \mid a^2+c^2=2b^2$, so $b$ is also even, giving a contradiction.
Thus $a$ is odd. Similarly $c$ is odd, so $b$ is also odd.
$$(a-b)(a+b)=a^2-b^2=b^2-c^2=(b-c)(b+c)$$
$$\frac{a-b}{2}\frac{a+b}{2}=\frac{b-c}{2}\frac{b+c}{2}$$
By factoring lemma, there exists integers $w, x, y, z$ such that $\frac{a-b}{2}=wx, \frac{a+b}{2}=yz, \frac{b-c}{2}=wy, \frac{b+c}{2}=xz$. Now
$$a=wx+yz, b=yz-wx=wy+xz, c=xz-wy$$
$y(z-w)=x(z+w)$. Again by factoring lemma, there exists integers $d, e, f, g$ such that $y=de, z-w=fg, x=df, z+w=eg$. Now
$$z=\frac{eg+fg}{2}=g\frac{e+f}{2}, w=\frac{eg-fg}{2}=g\frac{e-f}{2}$$ $$a=wx+yz=dfg\frac{e-f}{2}+deg\frac{e+f}{2}=\frac{dg}{2}(e^2+2ef-f^2)$$ $$b=wy+xz=deg\frac{e-f}{2}+dfg\frac{e+f}{2}=\frac{dg}{2}(e^2+f^2)$$ $$c=xz-wy=dfg\frac{e+f}{2}-deg\frac{e-f}{2}=\frac{dg}{2}(-e^2+2ef+f^2)$$
If $dg$ is divisible by $4$ or an odd prime $p$, then $a, b, c$ will not be relatively prime. Thus $dg=\pm 1$ or $\pm 2$.
If $dg=\pm 2$, then $a=\pm (e^2+2ef-f^2), b=\pm (e^2+f^2), c=\pm (-e^2+2ef+f^2)$. Clearly $\gcd(e, f) \mid \gcd(a, b, c)=1$, so $e, f$ are relatively prime. If $e, f$ have the same parity, then $a, b, c$ are all even, a contradiction, so $e, f$ have different parities.
If $dg= \pm 1$, then $a=\pm \frac{e^2+2ef-f^2}{2}, b=\pm \frac{e^2+f^2}{2}, c=\pm \frac{-e^2+2ef+f^2}{2}$. Thus $e, f$ must have the same parity. Put $e+f=2p, e-f=2q$, then $e=p+q, f=p-q$, and
$$a=\pm \frac{e^2+2ef-f^2}{2}=\pm \frac{(p+q)^2+2(p+q)(p-q)-(p-q)^2}{2}=\pm (p^2+2pq-q^2)$$ $$b=\pm \frac{e^2+f^2}{2}=\pm \frac{(p+q)^2+(p-q)^2}{2}=\pm (p^2+q^2)$$ $$c=\pm \frac{-e^2+2ef+f^2}{2}=\pm \frac{-(p+q)^2+2(p+q)(p-q)+(p-q)^2}{2}=\mp(-p^2+2pq+q^2)$$
Again, $p, q$ must be relatively prime and be of different parities.
Finally, combining the 2 cases, $a, b, c$ can be written as
$$a=\epsilon_a (p^2+2pq-q^2), b=\epsilon_b (p^2+q^2), c=\epsilon_c (-p^2+2pq+q^2)$$
where each $\epsilon_a, \epsilon_b, \epsilon_c$ are $1$ or $-1$.