Am I correct in saying denoting $F=(yy')^2 + \lambda y^2$. Thus the extrema corresponding to this problem are the extrema for the functional, $J[y] = \int_0^1 F dx.$
You are correct. If we wanna minimize a functional $J[y]$ subject to a constraint $H[y]=\text{const}$, then the new functional we want to minimize is $ J_{\lambda}[y]:= J[y]+\lambda H[y] = \int^1_0 \left((yy')^2+\lambda y^2\right) dx.\tag{1} $
Therefore they are solutions of the Euler-Lagrange equation $\lambda y -y^2y''-(yy')^2=0$ Could some clarify whether my Euler-Lagrange equation is correct.
Upon my checking, I think this is not correct, equation $(\star)$ in blue color is what I obtained.
Instead of plugging the formula, I still prefer the old school way of deriving the E-L equation, aka the first variation of the functional in (1) is zero when the extremum is attained at $y$: $ \lim_{\epsilon\to 0} \frac{d}{d\epsilon} J_{\lambda}[y+\epsilon v] = 0,\\ \text{for any smooth test function } v \text{ that does not change the boundary value of } y. $ This is to say $v=0$ on both end points.
Firstly computing $J_{\lambda}[y+\epsilon v]$ yields: $ \int^1_0 \left((y+\epsilon v)^2(y'+\epsilon v')^2+ \lambda (y+\epsilon v)^2\right) dx. $ Taking derivative w.r.t. $\epsilon$ yields: $ \int^1_0 \left(2(y+\epsilon v)v(y'+\epsilon v')^2+2(y+\epsilon v)^2 (y'+\epsilon v') v' + 2\lambda (y+\epsilon v) v\right) dx. $ Letting $\epsilon\to 0$ and setting the integral to be zero yield: $ \int^1_0 \left( y(y')^2v + \color{red}{y^2 y' v'} + \lambda y v\right) dx = 0. $ Integrating by parts on the red term and using $v=0$ on the end points: $ \int^1_0 \left( y(y')^2v - 2y(y')^2 v - y^2y''v + \lambda y v\right) dx = 0. $ By fundamental theorem of calculus of variation: $ \color{blue}{y(y')^2 + y^2y'' =\lambda y}.\tag{$\star$} $ Suppose $y\neq 0$ for any $x$, we have $ (y')^2 + y y'' =\lambda y \implies (y^2)'' = 2\lambda. $
Solving this yields: $ y =\pm \sqrt{\lambda x^2 + ax + b}. $ The constraint $\int^1_0 y^2\,dx = 3$, the boundary condition $y(0)=1$, $y(1)=2$ will pin down the parameters, and we reached the final answer: $ \boxed{y = \sqrt{-3x^2+6x+1}}. $